A349333 G.f. A(x) satisfies A(x) = 1 + x * A(x)^6 / (1 - x).

1, 1, 7, 64, 678, 7836, 95838, 1219527, 15979551, 214151601, 2921712145, 40444378948, 566634504256, 8019501351103, 114484746457075, 1646614155398872, 23837794992712680, 347081039681365623, 5079306905986689309, 74670702678690897079, 1102218694940440851877

Offset: 0

Author

Ilya Gutkovskiy, Nov 15 2021

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..836

Crossrefs

Cf. A002212, A307678, A349331, A349332, A349334, A349335.

Cf. A002295, A346648 (partial sums), A349362.

Programs

Maple

a:= n-> coeff(series(RootOf(1+x*A^6/(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]^6/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
Table[Sum[Binomial[n - 1, k - 1] Binomial[6 k, k]/(5 k + 1), {k, 0, n}], {n, 0, 20}]

PARI

{a(n) = my(A=[1]); for(m=1, n, A=concat(A, 0);
A[#A] = 1 + sum(k=1, m-1, (polcoeff(Ser(A)^6, k)) )); A[n+1]}
for(n=0, 30, print1(a(n), ", ")) \\ Vaclav Kotesovec, Nov 23 2024, after Paul D. Hanna

Formula

a(n) = Sum_{k=0..n} binomial(n-1,k-1) * binomial(6*k,k) / (5*k+1).

a(n) ~ 49781^(n + 1/2) / (72 * sqrt(3*Pi) * n^(3/2) * 5^(5*n + 3/2)). - Vaclav Kotesovec, Nov 15 2021