A382332 Expansion of 1/(1 - 4*x/(1-x)^2)^(7/2).

1, 14, 154, 1470, 12866, 106078, 837018, 6385262, 47420674, 344553902, 2458367898, 17272647966, 119770278978, 821068784382, 5572735854234, 37490757508302, 250247764120578, 1658681038111566, 10924592141535898, 71541334475749502, 466060971286552642

Offset

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..600

Formula

a(0) = 1; a(n) = 2 * Sum_{k=0..n-1} (7-5*k/n) * (n-k) * a(k).

a(n) = ((7*n+7)*a(n-1) - (7*n-28)*a(n-2) + (n-3)*a(n-3))/n for n > 2.

a(n) = Sum_{k=0..n} (-4)^k * binomial(-7/2,k) * binomial(n+k-1,n-k).

a(n) = 14*n*hypergeom([9/2, 1-n, 1+n], [3/2, 2], -1) for n > 0. - Stefano Spezia, Mar 30 2025

a(n) ~ 2^(5/4) * (1 + sqrt(2))^(2*n) * n^(5/2) / (15*sqrt(Pi)). - Vaclav Kotesovec, May 03 2025

Mathematica

Table[Sum[(-4)^k* Binomial[-7/2, k]*Binomial[n+k-1, n-k], {k, 0, n}], {n, 0, 30}] (* Vincenzo Librandi, May 12 2025 *)

Prog

(PARI) a(n) = sum(k=0, n, (-4)^k*binomial(-7/2, k)*binomial(n+k-1, n-k));
(Magma) R<x>:=PowerSeriesRing(Rationals(), 25); Coefficients(R!( 1/(1 - 4*x/(1-x)^2)^(7/2))); // Vincenzo Librandi, May 12 2025

Crossrefs

Cf. A110170, A377198, A382274.

Cf. A020918, A377200.

Sequence in context: A290675 A097227 A377196 * A229315 A257288 A125426

Adjacent sequences: A382329 A382330 A382331 * A382333 A382334 A382335

Keyword

nonn

Author

Seiichi Manyama, Mar 30 2025

Status

approved