0,5

Triangle of generalized binomial coefficients (n,k)_7; cf. A342889. - N. J. A. Sloane, Apr 03 2021

Seiichi Manyama, Rows n = 0..139, flattened

Johann Cigler, Pascal triangle, Hoggatt matrices, and analogous constructions, arXiv:2103.01652 [math.CO], 2021.

Johann Cigler, Some observations about Hoggatt triangles, Universität Wien (Austria, 2021).

T(n,m) = A142465(n,m)*binomial(n+6,m)/binomial(m+6,m).

Sum_{k=0..n} T(n, k) = A005365(n).

Triangle begins as:

1;

1, 1;

1, 8, 1;

1, 36, 36, 1;

1, 120, 540, 120, 1;

1, 330, 4950, 4950, 330, 1;

1, 792, 32670, 108900, 32670, 792, 1;

1, 1716, 169884, 1557270, 1557270, 169884, 1716, 1;

1, 3432, 736164, 16195608, 44537922, 16195608, 736164, 3432, 1;

1, 6435, 2760615, 131589315, 868489479, 868489479, 131589315, 2760615, 6435, 1;

T[n_, k_]:= Product[Binomial[n+j, k]/Binomial[k+j, k], {j, 0, 6}];

Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Nov 13 2022 *)

(PARI) T(n, k) = prod(j=0, 6, binomial(n+j, k)/binomial(k+j, k)); \\ Seiichi Manyama, Apr 01 2021

(Magma) [(&*[Binomial(n+j, k)/Binomial(k+j, k): j in [0..6]]): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 13 2022

(SageMath)

def A142467(n, k): return product(binomial(n+j, k)/binomial(k+j, k) for j in (0..6))

flatten([[A142467(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Nov 13 2022

Cf. A001263, A005365 (row sums), A056939, A056940, A056941.

Triangles of generalized binomial coefficients (n,k)_m (or generalized Pascal triangles) for m = 1,...,12: A007318 (Pascal), A001263, A056939, A056940, A056941, A142465, A142467, A142468, A174109, A342889, A342890, A342891.

nonn,easy,tabl

Roger L. Bagula, Sep 20 2008

Edited by the Associate Editors of the OEIS, May 17 2009

approved

0,2

Let V be the vector representation of SL(4) (of dimension 4) and let E be the exterior algebra of V (of dimension 16). Then a(n) is the dimension of the subspace of invariant tensors in the n-th tensor power of E. - Bruce Westbury, Feb 18 2021

This is the number of 4-vicious walkers (aka vicious 4-watermelons) - see Essam and Guttmann (1995). This is the 4-walker analog of A001181. - N. J. A. Sloane, Mar 22 2021

D. C. Fielder and C. O. Alford, "An investigation of sequences derived from Hoggatt sums and Hoggatt triangles", in G. E. Bergum et al., editors, Applications of Fibonacci Numbers: Proc. Third Internat. Conf. on Fibonacci Numbers and Their Applications, Pisa, Jul 25-29, 1988. Kluwer, Dordrecht, Vol. 3, 1990, pp. 77-88.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 0..845

J. W. Essam and A. J. Guttmann, Vicious walkers and directed polymer networks in general dimensions, Physical Review E, 52(6), (1995) pp. 5849-5862. See (60) and (63).

D. C. Fielder, Letter to N. J. A. Sloane, Jun 1988

D. C. Fielder and C. O. Alford, On a conjecture by Hoggatt with extensions to Hoggatt sums and Hoggatt triangles, Fib. Quart., 27 (1989), 160-168.

D. C. Fielder and C. O. Alford, An investigation of sequences derived from Hoggatt Sums and Hoggatt Triangles, Application of Fibonacci Numbers, 3 (1990) 77-88. Proceedings of 'The Third Annual Conference on Fibonacci Numbers and Their Applications,' Pisa, Italy, July 25-29, 1988. (Annotated scanned copy)

Nick Hobson, Python program for this sequence

Vaclav Kotesovec, Calculation of the asymptotic formula for the sequence A005366

From Richard L. Ollerton, Sep 12 2006: (Start)

a(n) = Hypergeometric4F3([-3-n, -2-n, -1-n, -n], [2, 3, 4], 1).

(n+3)*(n+4)*(n+5)*(n+6)*a(n) = 6*(n+1)*(n+3)*(n+4)*(2*n+5)*a(n-1) + 4*(n-1)*n*(4*n+7)*(4*n+9)*a(n-2); a(0)=1, a(1)=2. (End)

a(n) = S(4,n) where S(d,n) is defined in A005364. - Sean A. Irvine, May 29 2016

a(n) ~ 3 * 2^(4*n + 29/2) / (Pi^(3/2) * n^(15/2)). - Vaclav Kotesovec, Apr 01 2021

a := n -> hypergeom([-3-n, -2-n, -1-n, -n], [2, 3, 4], 1):

seq(simplify(a(n)), n=0..25); # Peter Luschny, Feb 18 2021

A005362[n_]:=HypergeometricPFQ[{-3-n, -2-n, -1-n, -n}, {2, 3, 4}, 1] (* Richard L. Ollerton, Sep 12 2006 *)

(Magma)

A056940:= func< n, k | (&*[Binomial(n+j, k)/Binomial(k+j, k): j in [0..3]]) >;

A005362:= func< n | (&+[A056940(n, k): k in [0..n]]) >;

[A005362(n): n in [0..30]]; // G. C. Greubel, Nov 14 2022

(SageMath)

def A005362(n): return simplify(hypergeometric([-3-n, -2-n, -1-n, -n], [2, 3, 4], 1))

[A005362(n) for n in range(41)] # G. C. Greubel, Nov 14 2022

Cf. A005364, A005365, A005366, A056940, A116925.

N. J. A. Sloane, Simon Plouffe

approved

0,2

Let V be the vector representation of SL(5) (of dimension 5) and let E be the exterior algebra of V (of dimension 32). Then a(n) is the dimension of the subspace of invariant tensors in the n-th tensor power of E. - Bruce Westbury, Feb 18 2021

This is the number of 5-vicious walkers (aka vicious 5-watermelons) - see Essam and Guttmann (1995). This is the 5-walker analog of A001181. - N. J. A. Sloane, Mar 27 2021

D. C. Fielder and C. O. Alford, "An investigation of sequences derived from Hoggatt sums and Hoggatt triangles", in G. E. Bergum et al., editors, Applications of Fibonacci Numbers: Proc. Third Internat. Conf. on Fibonacci Numbers and Their Applications, Pisa, Jul 25-29, 1988. Kluwer, Dordrecht, Vol. 3, 1990, pp. 77-88.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 0..681

J. W. Essam and A. J. Guttmann, Vicious walkers and directed polymer networks in general dimensions, Physical Review E, 52(6), (1995) pp. 5849-5862. See (60) and (63).

D. C. Fielder, Letter to N. J. A. Sloane, Jun 1988

D. C. Fielder and C. O. Alford, On a conjecture by Hoggatt with extensions to Hoggatt sums and Hoggatt triangles, Fib. Quart., 27 (1989), 160-168.

D. C. Fielder and C. O. Alford, An investigation of sequences derived from Hoggatt Sums and Hoggatt Triangles, Application of Fibonacci Numbers, 3 (1990) 77-88. Proceedings of 'The Third Annual Conference on Fibonacci Numbers and Their Applications,' Pisa, Italy, July 25-29, 1988. (Annotated scanned copy)

Vaclav Kotesovec, Calculation of the asymptotic formula for the sequence A005366

From Richard L. Ollerton, Sep 12 2006: (Start)

a(n) = Hypergeometric5F4([-4-n, -3-n, -2-n, -1-n, -n], [2,3,4,5], -1).

(n+4)*(n+5)^2*(n+6)*(n+7)*(n+8)*(252 +253*n +55*n^2)*a(n) = 3*(n+4)*(n+5)*(141120 + 362152*n + 373054*n^2 + 192647*n^3 + 52441*n^4 + 7161*n^5 + 385*n^6)*a(n-1) + n*(n-1)*(5738880 + 14311976*n + 14466242*n^2 + 7579175*n^3 + 2170343*n^4 + 322289*n^5 + 19415*n^6)*a(n-2) - 32*(n-1)^2*n^2*(n-2)*(n+1)*(560 + 363*n + 55*n^2)*a(n-3); a(-1)=a(0)=1, a(1)=2. (End)

a(n) = S(5,n) where S(d,n) is defined in A005364. - Sean A. Irvine, May 29 2016

a(n) ~ 9 * 2^(5*n + 27) / (sqrt(5) * Pi^2 * n^12). - Vaclav Kotesovec, Apr 01 2021

a(n) = Sum_{k=0..n} A056941(n, k) (row sums of triangle A056941). - G. C. Greubel, Nov 14 2022

a := n -> hypergeom([-4-n, -3-n, -2-n, -1-n, -n], [2, 3, 4, 5], -1):

seq(simplify(a(n)), n=0..25); # Peter Luschny, Feb 18 2021

# The following Maple program is based on Eq (60) of Essam-Guttmann (1995) and confirms that that sequence is the same as the present one. - N. J. A. Sloane, Mar 27 2021

v5 := proc(n) local t1, t2, t3, t4, t5;

if n=0 then 1

elif n=1 then 2

elif n=2 then 8

else

t1 := (4+n)*(5+n)^2*(6+n)*(7+n)*(8+n)*(252+253*n+55*n^2);

t2 := 3*(4+n)*(5+n)*(141120+362152*n + 373054*n^2+192647*n^3+52441*n^4 +7161*n^5 +385*n^6);

t3 := n*(1-n)*(5738880+14311976*n+14466242*n^2+7579175*n^3 +2170343*n^4+322289*n^5 + 19415*n^6);

t4 := 32*(2-n)*(1-n)^2*n^2*(1+n)*(560+363*n+55*n^2);

t5 := t2*v5(n-1)-t3*v5(n-2)+t4*v5(n-3);

t5/t1;

fi; end;

[seq(v5(n), n=0..20)];

A005363[n_]:=HypergeometricPFQ[{-4-n, -3-n, -2-n, -1-n, -n}, {2, 3, 4, 5}, -1] (* Richard L. Ollerton, Sep 12 2006 *)

(Magma)

A056941:= func< n, k | (&*[Binomial(n+j, k)/Binomial(k+j, k): j in [0..4]]) >;

A005363:= func< n | (&+[A056941(n, k): k in [0..n]]) >;

[A005363(n): n in [0..40]]; // G. C. Greubel, Nov 14 2022

(SageMath)

def A005363(n): return simplify(hypergeometric([-4-n, -3-n, -2-n, -1-n, -n], [2, 3, 4, 5], -1))

[A005363(n) for n in range(51)] # G. C. Greubel, Nov 14 2022

Cf. A000079, A000108, A001181, A005362, A005364, A005365, A005366, A116925.

Cf. A056941.

N. J. A. Sloane, Simon Plouffe

More terms from Sean A. Irvine, May 29 2016

approved

0,2

Let V be the vector representation of SL(6) (of dimension 6) and let E be the exterior algebra of V (of dimension 64). Then a(n) is the dimension of the subspace of invariant tensors in the n-th tensor power of E. - Bruce Westbury, Feb 03 2021

This is the number of 6-vicious walkers (aka vicious 6-watermelons) - see Essam and Guttmann (1995). This is the 6-walker analog of A001181. - N. J. A. Sloane, Mar 27 2021

D. C. Fielder and C. O. Alford, An investigation of sequences derived from Hoggatt sums and Hoggatt triangles, in G. E. Bergum et al., editors, Applications of Fibonacci Numbers: Proc. Third Internat. Conf. on Fibonacci Numbers and Their Applications, Pisa, Jul 25-29, 1988. Kluwer, Dordrecht, Vol. 3, 1990, pp. 77-88.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 0..573

J. W. Essam and A. J. Guttmann, Vicious walkers and directed polymer networks in general dimensions, Physical Review E, 52(6), (1995) pp. 5849-5862. See (60) and (63).

D. C. Fielder, Letter to N. J. A. Sloane, Jun 1988

D. C. Fielder and C. O. Alford, An investigation of sequences derived from Hoggatt Sums and Hoggatt Triangles, Application of Fibonacci Numbers, 3 (1990) 77-88. Proceedings of 'The Third Annual Conference on Fibonacci Numbers and Their Applications,' Pisa, Italy, July 25-29, 1988. (Annotated scanned copy)

Vaclav Kotesovec, Calculation of the asymptotic formula for the sequence A005366

a(n) = Hypergeometric6F5([-5-n, -4-n, -3-n, -2-n, -1-n, -n], [2, 3, 4, 5, 6], 1). - Richard L. Ollerton, Sep 13 2006

a(n) = S(6,n) where S(d,n) = 1 + Sum_{h=0..n-1} Product_{k=0..h} binomial(n+d-1-k,d) / binomial(d + k, d) [From Fielder and Alford]. - Sean A. Irvine, May 29 2016

a(n) ~ 135 * 2^(6*n + 40) / (sqrt(3) * Pi^(5/2) * n^(35/2)). - Vaclav Kotesovec, Apr 01 2021

A005364[n_]:=HypergeometricPFQ[{-5-n, -4-n, -3-n, -2-n, -1-n, -n}, {2, 3, 4, 5, 6}, 1] (* Richard L. Ollerton, Sep 13 2006 *)

(PARI) a(n) = my(d=6); 1 + sum(h=0, n-1, prod(k=0, h, binomial(n+d-1-k, d) / binomial(d + k, d))); \\ Michel Marcus, Feb 08 2021

(Magma)

A142465:= func< n, k | (&*[Binomial(n+j, k)/Binomial(k+j, k): j in [0..5]]) >;

A005364:= func< n | (&+[A142465(n, k): k in [0..n]]) >;

[A005364(n): n in [0..40]]; // G. C. Greubel, Nov 13 2022

(SageMath)

def A005364(n): return simplify(hypergeometric([-5-n, -4-n, -3-n, -2-n, -1-n, -n], [2, 3, 4, 5, 6], 1))

[A005364(n) for n in range(51)] # G. C. Greubel, Nov 13 2022

Row sums of A142465.

Cf. A000079, A000108, A001181, A005362, A005363, A005365, A005366, A116925.

More terms from Sean A. Irvine, May 29 2016

approved

0,2

Let V be the vector representation of SL(8) (of dimension 8) and let E be the exterior algebra of V (of dimension 256). Then a(n) is the dimension of the subspace of invariant tensors in the n-th tensor power of E. - Bruce Westbury, Feb 03 2021

This is the number of 8-vicious walkers (aka vicious 8-watermelons) - see Essam and Guttmann (1995). This is the 8-walker analog of A001181. - N. J. A. Sloane, Mar 27 2021

In general, for d > 0, a(n) ~ BarnesG(d+1) * 2^(d*n + (2*d+1)*(d-1)/2) / (sqrt(d) * Pi^((d-1)/2) * n^((d^2 - 1)/2)). - Vaclav Kotesovec, Apr 01 2021

D. C. Fielder and C. O. Alford, An investigation of sequences derived from Hoggatt sums and Hoggatt triangles, in G. E. Bergum et al., editors, Applications of Fibonacci Numbers: Proc. Third Internat. Conf. on Fibonacci Numbers and Their Applications, Pisa, Jul 25-29, 1988. Kluwer, Dordrecht, Vol. 3, 1990, pp. 77-88.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 0..438

J. W. Essam and A. J. Guttmann, Vicious walkers and directed polymer networks in general dimensions, Physical Review E, 52(6), (1995) pp. 5849-5862. See (60) and (63).

D. C. Fielder, Letter to N. J. A. Sloane, Jun 1988

D. C. Fielder and C. O. Alford, An investigation of sequences derived from Hoggatt Sums and Hoggatt Triangles, Application of Fibonacci Numbers, 3 (1990) 77-88. Proceedings of 'The Third Annual Conference on Fibonacci Numbers and Their Applications,' Pisa, Italy, July 25-29, 1988. (Annotated scanned copy)

Vaclav Kotesovec, Calculation of the asymptotic formula for the sequence A005366

a(n) = Hypergeometric8F7([-7-n, -6-n, -5-n, -4-n, -3-n, -2-n, -1-n, -n],[2, 3, 4, 5, 6, 7, 8], 1). - Richard L. Ollerton, Sep 13 2006

a(n) = S(8,n) where S(d,n) is defined in A005364. - Sean A. Irvine, May 29 2016

a(n) ~ 1913625 * 2^(8*n + 74) / (Pi^(7/2) * n^(63/2)). - Vaclav Kotesovec, Apr 01 2021

A005366[n_]:=HypergeometricPFQ[{-7-n, -6-n, -5-n, -4-n, -3-n, -2-n, -1-n, -n}, {2, 3, 4, 5, 6, 7, 8}, 1] (* Richard L. Ollerton, Sep 13 2006 *)

(PARI) a(n) = my(d=8); 1 + sum(h=0, n-1, prod(k=0, h, binomial(n+d-1-k, d) / binomial(d + k, d))); \\ Michel Marcus, Feb 08 2021

(Magma)

A142468:= func< n, k | Binomial(n, k)*(&*[Binomial(n+2*j, k+j)/Binomial(n+2*j, j): j in [1..7]]) >;

A005366:= func< n | (&+[A142468(n, k): k in [0..n]]) >;

[A005366(n): n in [0..40]]; // G. C. Greubel, Nov 13 2022

(SageMath)

def A005365(n): return simplify(hypergeometric([-7-n, -6-n, -5-n, -4-n, -3-n, -2-n, -1-n, -n], [2, 3, 4, 5, 6, 7, 8], 1))

[A005365(n) for n in range(51)] # G. C. Greubel, Nov 13 2022

Row sums of A142468.

Cf. A000079, A000108, A001181, A005362, A005363, A005364, A005365, A116925.

More terms from Sean A. Irvine, May 29 2016

approved

0,2

Seiichi Manyama, Table of n, a(n) for n = 0..87

a(n) = Sum_{j=0..n} Product_{k=0..n-1} binomial(n+k,j)/binomial(j+k,j).

a(n) ~ c * exp(1/12) * 2^(4*n^2 - 1/12) / (A * n^(1/12) * 3^(9*n^2/4 - 1/6)), where c = JacobiTheta3(0,1/3) = EllipticTheta[3, 0, 1/3] = 1.69145968168171534134842... if n is even, and c = JacobiTheta2(0,1/3) = EllipticTheta[2, 0, 1/3] = 1.69061120307521423305296... if n is odd, and A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Apr 02 2021

a[n_] := 1 + Sum[Product[Binomial[2*n - 1, n + k]/Binomial[2*n - 1, k], {k, 0, j - 1}], {j, 1, n}]; Array[a, 17, 0] (* Amiram Eldar, Apr 01 2021 *)

Table[1 + BarnesG[2*n + 1] * Sum[BarnesG[j + 1]*BarnesG[n - j + 1] / (BarnesG[n + j + 1]*BarnesG[2*n - j + 1]), {j, 1, n}], {n, 0, 15}] (* Vaclav Kotesovec, Apr 02 2021 *)

(PARI) a(n) = 1+sum(j=1, n, prod(k=0, j-1, binomial(2*n-1, n+k)/binomial(2*n-1, k)));

(PARI) a(n) = sum(j=0, n, prod(k=0, n-1, binomial(n+k, j)/binomial(j+k, j)));

Row sums of A342972.

Cf. A001181, A005362, A005363, A005364, A005365, A005366, A116925.

Seiichi Manyama, Apr 01 2021

approved