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
Keywords
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Programs
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Magma
R
:=PowerSeriesRing(Rationals(), 35); Coefficients(R!( 1/(1 - 4*x/(1-x))^(7/2))); // Vincenzo Librandi, May 11 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 *) -
PARI
a(n) = sum(k=0, n, (-4)^k*binomial(-7/2, k)*binomial(n-1, n-k));
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