A391099 Expansion of g/(1 - x^2*g^3), where g = 1+x*g^4 is the g.f. of A002293.

1, 1, 5, 26, 163, 1116, 8103, 61271, 477333, 3804635, 30878286, 254306304, 2119951278, 17853490116, 151669244490, 1298170716509, 11184287356123, 96913643138408, 844068755391993, 7384974708096903, 64877915485235577, 572070036121210437, 5061247928589279937

Offset

0,3

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Formula

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

Mathematica

Table[Sum[ (3*k+1)*Binomial[4* n-5*k+1, n-2*k]/(4*n-5*k+1), {k, 0, Floor[n/2]}], {n, 0, 26}] (* Vincenzo Librandi, Nov 30 2025 *)

Prog

(PARI) a(n) = sum(k=0, n\2, (3*k+1)*binomial(4*n-5*k+1, n-2*k)/(4*n-5*k+1));
(Magma) [&+[(3*k+1)*Binomial(4*n-5*k+1, n-2*k)/(4*n-5*k+1): k in [0..Floor(n/2)]] : n in [0..40] ]; // Vincenzo Librandi, Nov 30 2025

Crossrefs

Cf. A109963, A390280, A391097, A391104.

Cf. A002293, A391080.

Sequence in context: A302442 A367285 A349882 * A209672 A367041 A057793

Adjacent sequences: A391096 A391097 A391098 * A391100 A391101 A391102

Keyword

nonn

Author

Seiichi Manyama, Nov 28 2025

Status

approved