A061676 - OEIS

A061676

Triangle T(n,k) of number of ways of throwing k standard dice to produce a total of n.

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, 1, 5, 10, 10, 5, 1, 0, 6, 15, 20, 15, 6, 1, 0, 5, 21, 35, 35, 21, 7, 1, 0, 4, 25, 56, 70, 56, 28, 8, 1, 0, 3, 27, 80, 126, 126, 84, 36, 9, 1, 0, 2, 27, 104, 205, 252, 210, 120, 45, 10, 1, 0, 1, 25, 125, 305, 456, 462, 330, 165, 55, 11, 1

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OFFSET

1,5

LINKS

Table of n, a(n) for n=1..78.

FORMULA

T(n, k)=T(n-1, k-1)+T(n-2, k-1)+T(n-3, k-1)+T(n-4, k-1)+T(n-5, k-1)+T(n-6, k-1) starting with T(0, 0)=1. T(n, k)=T(7k-n, k); if n>6k or n<k, T(n, k)=0; if n<k+6, T(n, k)=C(n-1, k-1); if n>6k-6, T(n, k)=C(7k-n-1, k-1); T([7k/2], k)=A018901(k).

EXAMPLE

Rows start:

1,1;

1,2,1;

1,3,3,1;

1,4,6,4,1;

1,5,10,10,5,1;

0,6,15,20,15,6,1;

0,5,21,35,35,21,7,1;

etc.

T(8,2)=5 since 8 =2+6 =3+5 =4+4 =5+3 =6+2.

MAPLE

pts := 6; # A213889 and A061676

g := 1/(1-t*z*add(z^i, i=0..pts-1)) ;

for n from 1 to 13 do

for k from 1 to n do

coeftayl(g, z=0, n) ;

coeftayl(%, t=0, k) ;

printf("%d ", %) ;

end do:

printf("\n") ;

end do: # R. J. Mathar, May 28 2025

CROSSREFS

First 21 terms as A007318 (see formula). Cf. A001592, A069713.

Cf. A030528 (2-sided dice), A078803 (3-sided), A213887 (4-sided), A213888 (5-sided).

Sequence in context: A089239 A223968 A214846 * A180182 A275198 A095145

Adjacent sequences: A061673 A061674 A061675 * A061677 A061678 A061679

KEYWORD

nonn,tabl

AUTHOR

Henry Bottomley, Apr 01 2002

STATUS

approved