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A374508
Expansion of 1/(1 - 2*x + 5*x^2)^(5/2).
1
1, 5, 5, -35, -140, -84, 840, 2640, 495, -16445, -41041, 11375, 282100, 559300, -474300, -4399260, -6807225, 11062275, 63677075, 73363675, -208411280, -865816600, -665544100, 3475847700, 11129861925, 4130560161, -53332660395, -135538728395, 9634906640
OFFSET
0,2
FORMULA
a(0) = 1, a(1) = 5; a(n) = ((2*n+3)*a(n-1) - 5*(n+3)*a(n-2))/n.
a(n) = (binomial(n+4,2)/6) * Sum_{k=0..floor(n/2)} (-1)^k * binomial(n+2,n-2*k) * binomial(2*k+2,k).
a(n) = Pochhammer(n+1, 4)*hypergeom([(1-n)/2, -n/2], [3], -4)/4!. - Stefano Spezia, Jul 10 2024
MATHEMATICA
a[n_]:= Pochhammer[n+1, 4]*Hypergeometric2F1[(1-n)/2, -n/2, 3, -4]/4!; Array[a, 29, 0] (* Stefano Spezia, Jul 10 2024 *)
PROG
(PARI) a(n) = binomial(n+4, 2)/6*sum(k=0, n\2, (-1)^k*binomial(n+2, n-2*k)*binomial(2*k+2, k));
CROSSREFS
KEYWORD
sign
AUTHOR
Seiichi Manyama, Jul 09 2024
STATUS
approved