A391456 - OEIS

%I #11 Dec 20 2025 13:06:23

%S 1,2,9,51,336,2372,17571,134757,1060967,8526011,69649968,576686912,

%T 4828760456,40818936354,347881337345,2985889744311,25787373559671,

%U 223931861330691,1954056716589431,17125740608096079,150682919553817883,1330514274189243211

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

%H Vincenzo Librandi, <a href="/A391456/b391456.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) = Sum_{k=0..floor(n/3)} (-1)^k * (k+1) * binomial(4*n-10*k+2,n-3*k)/(2*n-5*k+1).

%t Table[Sum[(-1)^k*(k+1)*Binomial[4*n-10*k+2,n-3*k]/(2*n-5*k+1),{k,0,Floor[n/3]}],{n,0,25}] (* _Vincenzo Librandi_, Dec 20 2025 *)

%o (PARI) a(n) = sum(k=0, n\3, (-1)^k*(k+1)*binomial(4*n-10*k+2, n-3*k)/(2*n-5*k+1));

%o (Magma) [&+[(-1)^k*(k+1)*Binomial(4*n-10*k+2, n-3*k)/(2*n-5*k+1): k in [0..Floor(n/3)]] : n in [0..25] ]; //  _Vincenzo Librandi_, Dec 20 2025

%Y Cf. A002293, A391079.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Dec 10 2025