A389115 - OEIS

%I #90 Dec 09 2025 08:17:01

%S 1,1,4,11,35,113,376,1276,4402,15390,54408,194169,698567,2530937,

%T 9226160,33815432,124538318,460644494,1710476504,6373775030,

%U 23826843614,89331823706,335823046544,1265571538136,4780273605460,18094046107408,68623099425856,260734318876725

%N Expansion of g^3/(1 + x*g)^2, where g = 1+x*g^2 is the g.f. of A000108.

%F a(n) = Sum_{k=0..n} (-1)^k * (k+1) * (k+3) * binomial(2*n-k+3,n-k)/(2*n-k+3).

%F a(n) = (1/(n+3)) * Sum_{k=0..n} (-1)^k * (k+1) * (k+3) * binomial(2*n-k+2,n-k).

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

%Y Cf. A065601, A114495, A391408.

%Y Cf. A104629, A294527.

%Y Cf. A000108, A000344, A391405.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Dec 08 2025