OFFSET
0,2
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..1000
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
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 *)
PROG
(PARI) my(N=30, x='x+O('x^N)); Vec(1/sqrt(1-4*x/(1+x)^6))
CROSSREFS
KEYWORD
sign
AUTHOR
Seiichi Manyama, Mar 24 2023
STATUS
approved