A377200 Expansion of 1/(1 - 4*x/(1-x))^(7/2).

1, 14, 140, 1190, 9170, 66122, 454328, 3009050, 19359620, 121664410, 749879508, 4546925922, 27188341530, 160624341990, 939009926520, 5438826037974, 31244200818306, 178173537480330, 1009366349014100, 5684102310204850, 31836106214747590, 177430881586034110

Offset

0,2

Links

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

Formula

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

a(n) = (2*(3*n+4)*a(n-1) - 5*(n-2)*a(n-2))/n for n > 1.

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

a(n) ~ 1024 * 5^(n - 9/2) * n^(5/2) / (3*sqrt(Pi)). - Vaclav Kotesovec, May 03 2025

a(n) = 14*hypergeom([9/2, 1-n], [2], -4) for n > 0. - Stefano Spezia, May 08 2025

Mathematica

Table[Sum[(-4)^k*Binomial[-7/2, k]*Binomial[n-1, n-k], {k, 0, n}], {n, 0, 30}] (* Vincenzo Librandi, May 11 2025 *)

Prog

(PARI) a(n) = sum(k=0, n, (-4)^k*binomial(-7/2, k)*binomial(n-1, n-k));
(Magma) R<x>:=PowerSeriesRing(Rationals(), 35); Coefficients(R!( 1/(1 - 4*x/(1-x))^(7/2))); // Vincenzo Librandi, May 11 2025

Crossrefs

Cf. A085362, A377197, A377199.

Sequence in context: A338323 A121034 A374513 * A125402 A016290 A003457

Adjacent sequences: A377197 A377198 A377199 * A377201 A377202 A377203

Keyword

nonn

Author

Seiichi Manyama, Oct 19 2024

Status

approved