cp's OEIS Frontend

A061676 Triangle T(n,k) of number of ways of throwing k standard dice to produce a total of n.

%I A061676 #8 May 28 2025 04:53:07
%S A061676 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,
%T A061676 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,
%U A061676 27,104,205,252,210,120,45,10,1,0,1,25,125,305,456,462,330,165,55,11,1
%N A061676 Triangle T(n,k) of number of ways of throwing k standard dice to produce a total of n.
%F A061676 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).
%e A061676 Rows start:
%e A061676 1;
%e A061676 1,1;
%e A061676 1,2,1;
%e A061676 1,3,3,1;
%e A061676 1,4,6,4,1;
%e A061676 1,5,10,10,5,1;
%e A061676 0,6,15,20,15,6,1;
%e A061676 0,5,21,35,35,21,7,1;
%e A061676 etc.
%e A061676 T(8,2)=5 since 8 =2+6 =3+5 =4+4 =5+3 =6+2.
%p A061676 pts := 6; # A213889 and A061676
%p A061676 g := 1/(1-t*z*add(z^i,i=0..pts-1)) ;
%p A061676 for n from 1 to 13 do
%p A061676     for k from 1 to n do
%p A061676         coeftayl(g,z=0,n) ;
%p A061676         coeftayl(%,t=0,k) ;
%p A061676         printf("%d ",%) ;
%p A061676     end do:
%p A061676     printf("\n") ;
%p A061676 end do: # _R. J. Mathar_, May 28 2025
%Y A061676 First 21 terms as A007318 (see formula). Cf. A001592, A069713.
%Y A061676 Cf. A030528 (2-sided dice), A078803 (3-sided), A213887 (4-sided), A213888 (5-sided).
%K A061676 nonn,tabl
%O A061676 1,5
%A A061676 _Henry Bottomley_, Apr 01 2002