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
Keywords
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..600
Programs
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Magma
R
:=PowerSeriesRing(Rationals(), 25); Coefficients(R!( 1/(1 - 4*x/(1-x)^2)^(7/2))); // Vincenzo Librandi, May 12 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 *) -
PARI
a(n) = sum(k=0, n, (-4)^k*binomial(-7/2, k)*binomial(n+k-1, n-k));
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