A391609 - OEIS

%I #13 Dec 18 2025 01:22:07

%S 1,0,6,33,231,1680,12722,99174,790695,6417782,52854210,440569659,

%T 3709925170,31512765744,269690266146,2323196695551,20128333751349,

%U 175286209221924,1533444732404226,13469998501882377,118760989817062512,1050602774154917980,9322513339131716088

%N Expansion of (g/(1 + x*g^2))^3, where g = 1+x*g^4 is the g.f. of A002293.

%H Vincenzo Librandi, <a href="/A391609/b391609.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f.: B(x)^3, where B(x) is the g.f. of A390740.

%F a(n) = Sum_{k=0..n} (-1)^k * (2*k+3) * binomial(k+2,2) * binomial(4*n-2*k+3,n-k)/(4*n-2*k+3).

%t Table[Sum[(-1)^k*(2*k+3)*Binomial[k+2,2]*Binomial[4*n-2*k+3,n-k]/(4*n-2*k+3),{k,0,n}],{n,0,25}] (* _Vincenzo Librandi_, Dec 17 2025 *)

%o (PARI) a(n) = sum(k=0, n, (-1)^k*(2*k+3)*binomial(k+2, 2)*binomial(4*n-2*k+3, n-k)/(4*n-2*k+3));

%o (Magma) [&+[(-1)^k*(2*k+3)*Binomial(k+2,2) * Binomial(4*n-2*k+3, n-k)/(4*n-2*k+3): k in [0..n]] : n in [0..30] ]; //  _Vincenzo Librandi_, Dec 17 2025

%Y Cf. A002293, A390740, A391202.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Dec 14 2025