A382274 Expansion of 1/(1 - 4*x/(1-x)^2)^(5/2).
1, 10, 90, 730, 5570, 40762, 289370, 2007210, 13671170, 91750250, 608294490, 3991833210, 25968131010, 167664187290, 1075453670490, 6858654320970, 43517809896450, 274862176368330, 1728960219827290, 10835520927931930, 67679638209628098, 421442759107879930
Offset: 0
Keywords
Programs
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PARI
a(n) = sum(k=0, n, (-4)^k*binomial(-5/2, k)*binomial(n+k-1, n-k));
Formula
a(0) = 1; a(n) = 2 * Sum_{k=0..n-1} (5-3*k/n) * (n-k) * a(k).
a(n) = ((7*n+3)*a(n-1) - (7*n-24)*a(n-2) + (n-3)*a(n-3))/n for n > 2.
a(n) = Sum_{k=0..n} (-4)^k * binomial(-5/2,k) * binomial(n+k-1,n-k).
a(n) = 10*n*hypergeom([7/2, 1-n, 1+n], [3/2, 2], -1) for n > 0. - Stefano Spezia, Mar 30 2025
a(n) ~ 2^(3/4) * n^(3/2) * (1 + sqrt(2))^(2*n) / (3*sqrt(Pi)). - Vaclav Kotesovec, Apr 13 2025