A361792 Expansion of 1/sqrt(1 - 4*x/(1+x)^6).
1, 2, -6, -10, 66, 60, -750, -236, 8682, -2098, -100792, 80286, 1162458, -1603412, -13225764, 26767020, 147428498, -409582818, -1596563202, 5941802122, 16587101544, -83014131140, -161717252990, 1126247965980, 1411774064970, -14905602076350
Offset: 0
Keywords
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..1000
Programs
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Mathematica
a[n_]:=(-1)^(n+1)Pochhammer[n,5]HypergeometricPFQ[{1-n,1+n/5,(6+n)/5, (7+n)/5, (8+n)/5, (9+n)/5}, {7/6,4/3,5/3,11/6,2}, 5^5/(2^4*3^6)]/60; Join[{1},Array[a,25]] (* Stefano Spezia, Jul 11 2024 *) -
PARI
my(N=30, x='x+O('x^N)); Vec(1/sqrt(1-4*x/(1+x)^6))
Formula
a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(2*k,k) * binomial(n+5*k-1,n-k).
n*a(n) = -( (3*n-5)*a(n-1) + (17*n-24)*a(n-2) + 35*(n-3)*a(n-3) + 35*(n-4)*a(n-4) + 21*(n-5)*a(n-5) + 7*(n-6)*a(n-6) + (n-7)*a(n-7) ) for n > 6.
a(0) = 1; a(n) = (2/n) * Sum_{k=0..n-1} (-1)^(n-1-k) * (n+k) * binomial(n+4-k,5) * a(k).
a(n) = (-1)^(n+1)*Pochhammer(n,5)*hypergeom([1-n, 1+n/5, (6+n)/5, (7+n)/5, (8+n)/5, (9+n)/5], [7/6, 4/3, 5/3, 11/6, 2], 5^5/(2^4*3^6))/60 for n > 0. - Stefano Spezia, Jul 11 2024