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A361791
Expansion of 1/sqrt(1 - 4*x/(1+x)^5).
7
1, 2, -4, -10, 30, 72, -238, -580, 1970, 4910, -16734, -42750, 144600, 379000, -1264700, -3402480, 11160730, 30828070, -99168820, -281279030, 885931600, 2580541580, -7948885910, -23779051760, 71572652480, 219906488302, -646332447086, -2039738985238, 5850898295170
OFFSET
0,2
LINKS
FORMULA
a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(2*k,k) * binomial(n+4*k-1,n-k).
n*a(n) = -( (2*n-4)*a(n-1) + (11*n-14)*a(n-2) + 20*(n-3)*a(n-3) + 15*(n-4)*a(n-4) + 6*(n-5)*a(n-5) + (n-6)*a(n-6) ) for n > 5.
a(0) = 1; a(n) = (2/n) * Sum_{k=0..n-1} (-1)^(n-1-k) * (n+k) * binomial(n+3-k,4) * a(k).
a(n) = (-1)^(n+1)*Pochhammer(n,4)*hypergeom([3/2, 1-n, 1+n/4, (5+n)/4, (6+n)/4, (7+n)/4], [6/5, 7/5, 8/5, 9/5, 2], 2^10/5^5)/12 for n > 0. - Stefano Spezia, Jul 11 2024
MATHEMATICA
a[n_]:=(-1)^(n+1)Pochhammer[n, 4]HypergeometricPFQ[{3/2, 1-n, 1+n/4, (5+n)/4, (6+n)/4, (7+n)/4}, {6/5, 7/5, 8/5, 9/5, 2}, 2^10/5^5]/12; Join[{1}, Array[a, 28]] (* Stefano Spezia, Jul 11 2024 *)
PROG
(PARI) my(N=30, x='x+O('x^N)); Vec(1/sqrt(1-4*x/(1+x)^5))
(PARI) a(n) = sum(k=0, n, (-1)^(n-k) * binomial(2*k, k) * binomial(n+4*k-1, n-k)) \\ Winston de Greef, Mar 24 2023
CROSSREFS
KEYWORD
sign
AUTHOR
Seiichi Manyama, Mar 24 2023
STATUS
approved