A377200 - OEIS

%I #22 May 11 2025 09:26:05

%S 1,14,140,1190,9170,66122,454328,3009050,19359620,121664410,749879508,

%T 4546925922,27188341530,160624341990,939009926520,5438826037974,

%U 31244200818306,178173537480330,1009366349014100,5684102310204850,31836106214747590,177430881586034110

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

%H Vincenzo Librandi, <a href="/A377200/b377200.txt">Table of n, a(n) for n = 0..1000</a>

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

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

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

%F a(n) ~ 1024 * 5^(n - 9/2) * n^(5/2) / (3*sqrt(Pi)). - _Vaclav Kotesovec_, May 03 2025

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

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

%o (PARI) a(n) = sum(k=0, n, (-4)^k*binomial(-7/2, k)*binomial(n-1, n-k));

%o (Magma) R<x>:=PowerSeriesRing(Rationals(), 35); Coefficients(R!( 1/(1 - 4*x/(1-x))^(7/2))); // _Vincenzo Librandi_, May 11 2025

%Y Cf. A085362, A377197, A377199.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Oct 19 2024