A125277 - OEIS

%I #3 Mar 30 2012 18:37:01

%S 1,1,2,7,32,169,981,6113,40386,280871,2047316,15595317,123876270,

%T 1024188790,8799533250,78443220865,724472766347,6922133112818,

%U 68331103658847,695983854400857,7305630631254242,78941171881894716

%N Eigensequence of triangle A110616: a(n) = Sum_{k=0..n-1} A110616(n-1,k)*a(k) for n>0 with a(0)=1.

%F a(n) = Sum_{k=0..n-1} a(k) * C(3*n-2*k-2,n-k-1)*(k+1)/(3*n-2*k-2) for n>0 with a(0)=1.

%e a(3) = 3*(1) + 2*(1) + 1*(2) = 7;

%e a(4) = 12*(1) + 7*(1) + 3*(2) + 1*(7) = 32;

%e a(5) = 55*(1) + 30*(1) + 12*(2) + 4*(7) + 1*(32) = 169.

%e Triangle A110616(n,k) = C(3*n-2*k+1, n-k)*(k+1)/(3*n-2*k+1) begins:

%e 1;

%e 1, 1;

%e 3, 2, 1;

%e 12, 7, 3, 1;

%e 55, 30, 12, 4, 1;

%e 273, 143, 55, 18, 5, 1;

%e 1428, 728, 273, 88, 25, 6, 1; ...

%e where g.f. of column k = G001764(x)^(k+1)

%e and G001764(x) = 1 + x*G001764(x)^3 is the g.f. of A001764.

%o (PARI) {a(n)=if(n==0,1,sum(k=0,n-1, a(k)*binomial(3*n-2*k-2,n-k-1)*(k+1)/(3*n-2*k-2)))}

%Y Cf. A110616, A001764, A091768.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Nov 26 2006