A349364 G.f. A(x) satisfies: A(x) = 1 + x * A(x)^8 / (1 + x).

1, 1, 7, 77, 987, 13839, 205513, 3176747, 50578445, 823779286, 13660621282, 229865812134, 3915003083306, 67361559577578, 1169138502393414, 20444573270374050, 359858503314494318, 6370677542063831319, 113359050598950194801, 2026309136822686950087

Offset: 0

Author

Ilya Gutkovskiy, Nov 15 2021

Keywords

• nonn

Comments

In general, for m > 1, Sum_{k=0..n} (-1)^(n-k) * binomial(n-1,k-1) * binomial(m*k,k) / ((m-1)*k+1) ~ (m-1)^(m/2 - 2) * (m^m/(m-1)^(m-1) - 1)^(n + 1/2) / (sqrt(2*Pi) * m^((m-1)/2) * n^(3/2)). - Vaclav Kotesovec, Nov 17 2021

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..500

Crossrefs

Cf. A001006, A007556, A127897, A317133, A346668, A346672 (binomial transform), A349335, A349361, A349362, A349363.

Programs

Maple

a:= n-> coeff(series(RootOf(1+x*A^8/(1+x)-A, A), x, n+1), x, n):
seq(a(n), n=0..19);  # Alois P. Heinz, Nov 15 2021

Mathematica

nmax = 19; A[] = 0; Do[A[x] = 1 + x A[x]^8/(1 + x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
Table[Sum[(-1)^(n - k) Binomial[n - 1, k - 1] Binomial[8 k, k]/(7 k + 1), {k, 0, n}], {n, 0, 19}]

Formula

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n-1,k-1) * binomial(8*k,k) / (7*k+1).

a(n) = (-1)^(n+1)*F([9/8, 5/4, 11/8, 3/2, 13/8, 7/4, 15/8, 1-n], [9/7, 10/7, 11/7, 12/7, 13/7, 2, 15/7], 8^8/7^7), where F is the generalized hypergeometric function. - Stefano Spezia, Nov 15 2021

a(n) ~ 15953673^(n + 1/2) / (2048 * sqrt(Pi) * n^(3/2) * 7^(7*n + 3/2)). - Vaclav Kotesovec, Nov 17 2021