OFFSET
0,3
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..500
FORMULA
a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n-1,k-1) * binomial(5*k,k) / (4*k+1).
a(n) = (-1)^(n+1)*F([6/5, 7/5, 8/5, 9/5, 1-n], [3/2, 7/4, 2, 9/4], 5^5/2^8), where F is the generalized hypergeometric function. - Stefano Spezia, Nov 15 2021
From Vaclav Kotesovec, Nov 17 2021: (Start)
a(n) ~ 2869^(n + 1/2) / (25 * sqrt(Pi) * n^(3/2) * 2^(8*n + 7/2)).
Recurrence: 8*n*(2*n - 1)*(4*n - 1)*(4*n + 1)*a(n) = 3*(615*n^4 - 718*n^3 - 275*n^2 + 618*n - 200)*a(n-1) + 4*(n-2)*(2485*n^3 - 6879*n^2 + 6524*n - 2040)*a(n-2) + 2*(n-3)*(n-2)*(8095*n^2 - 23517*n + 18092)*a(n-3) + 12*(n-4)*(n-3)*(n-2)*(935*n - 1838)*a(n-4) + 2869*(n-5)*(n-4)*(n-3)*(n-2)*a(n-5). (End)
MAPLE
a:= n-> coeff(series(RootOf(1+x*A^5/(1+x)-A, A), x, n+1), x, n):
seq(a(n), n=0..20); # Alois P. Heinz, Nov 15 2021
MATHEMATICA
nmax = 20; A[_] = 0; Do[A[x_] = 1 + x A[x]^5/(1 + x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
Table[Sum[(-1)^(n - k) Binomial[n - 1, k - 1] Binomial[5 k, k]/(4 k + 1), {k, 0, n}], {n, 0, 20}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Nov 15 2021
STATUS
approved