A344399 a(n) = 4^n*binomial(n - 1/2, -1/2)*(n^2 + 1).

1, 4, 30, 200, 1190, 6552, 34188, 171600, 836550, 3986840, 18660356, 86062704, 392102620, 1768102000, 7902970200, 35056559520, 154477660230, 676745803800, 2949418972500, 12794985495600, 55276458056820, 237909980502480, 1020487997404200, 4363718285628000

Offset: 0

Author

Peter Luschny, May 17 2021

Keywords

• nonn
• changed

Crossrefs

Cf. A344400, A344401.

Programs

Maple

aList := proc(len) local gf, ser;
gf := (20*x^2 - 6*x + 1) / (1 - 4*x)^(5/2): ser := series(gf, x, len+2):
seq(coeff(ser, x, n), n = 0..len) end: aList(23);

Mathematica

Table[4^n Binomial[n-1/2,-1/2](n^2+1),{n,0,30}] (* Harvey P. Dale, Jun 20 2021 *)

Formula

a(n) = [x^n] (20*x^2 - 6*x + 1) / (1 - 4*x)^(5/2).

a(n) = a(n-1)*(-2 + 4*n - 2*n^2 + 4*n^3) / (2*n - 2*n^2 + n^3) for n > 0.

a(n) ~ 4^n * n^(3/2) / sqrt(Pi). - Amiram Eldar, Sep 07 2025