A110511 - OEIS

A110511

Riordan array (1/(1+x), x(1-x)/(1+x)^2).

1, -1, 1, 1, -4, 1, -1, 9, -7, 1, 1, -16, 26, -10, 1, -1, 25, -70, 52, -13, 1, 1, -36, 155, -190, 87, -16, 1, -1, 49, -301, 553, -403, 131, -19, 1, 1, -64, 532, -1372, 1462, -736, 184, -22, 1, -1, 81, -876, 3024, -4446, 3206, -1216, 246, -25, 1, 1, -100, 1365, -6084, 11826, -11584, 6190, -1870, 317, -28, 1

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OFFSET

0,5

COMMENTS

Inverse of number triangle A110506. Row sums are A110512. Diagonal sums are A110513. Product of (1/(1+x), x/(1+x)) (inverse binomial transform matrix) and (1, x(1-2x)) (A110509).

LINKS

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

FORMULA

Number triangle: T(n, k) = Sum_{j=0..n} (-1)^(n-j)*C(n, j)*(-2)^(j-k)*C(k, j-k).

T(n, k) = Sum_{j=0..n} Sum_{i=0..k} C(k, i)*C(n+k-i-j-1, n-k-i-j)*(-1)^(n-k).

T(n,k) = T(n-1,k-1) - 2*T(n-1,k) - T(n-2,k) - T(n-2,k-1), T(0,0)=1, T(1,0)=-1, T(1,1)=1, T(n,k)=0 if k < 0 or if k > n. - Philippe Deléham, Jan 12 2014

EXAMPLE

Rows begin

-1, 1;

1, -4, 1;

-1, 9, -7, 1;

1, -16, 26, -10, 1;

-1, 25, -70, 52, -13, 1;

MATHEMATICA

T[n_, k_] := Sum[(-1)^(n - j)*Binomial[n, j]*(-2)^(j - k)*Binomial[k, j - k], {j, 0, n}]; Table[T[n, k], {n, 0, 20}, {k, 0, n}] // Flatten (* G. C. Greubel, Aug 29 2017 *)

PROG

(PARI) for(n=0, 20, for(k=0, n, print1(sum(j=0, n, (-1)^(n-j)*binomial(n, j)*(-2)^(j-k)*binomial(k, j-k)), ", "))) \\ G. C. Greubel, Aug 29 2017

CROSSREFS

Cf. A110506, A110512, A110513, A110509, A114188,

Sequence in context: A183153 A208513 A141905 * A114188 A082950 A060102

Adjacent sequences: A110508 A110509 A110510 * A110512 A110513 A110514

KEYWORD

easy,sign,tabl

AUTHOR

Paul Barry, Jul 24 2005

STATUS

approved