A349332 G.f. A(x) satisfies A(x) = 1 + x * A(x)^5 / (1 - x).

1, 1, 6, 46, 406, 3901, 39627, 418592, 4551672, 50610692, 572807157, 6577068383, 76426719408, 897078662538, 10620634999318, 126676885170703, 1520759193166329, 18361269213121164, 222814883564042704, 2716125963857227904, 33244557641365865109

Offset: 0

Author

Ilya Gutkovskiy, Nov 15 2021

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..897

Crossrefs

Cf. A002212, A307678, A349331, A349333, A349334, A349335.

Cf. A002294, A346647 (partial sums), A349361, A364748.

Programs

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[Binomial[n - 1, k - 1] Binomial[5 k, k]/(4 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)^5, 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(5*k,k) / (4*k+1).

a(n) ~ 3381^(n + 1/2) / (25 * sqrt(Pi) * n^(3/2) * 2^(8*n + 7/2)). - Vaclav Kotesovec, Nov 15 2021

Recurrence: 8*n*(2*n - 1)*(4*n - 1)*(4*n + 1)*a(n) = (4405*n^4 - 10346*n^3 + 9575*n^2 - 4354*n + 840)*a(n-1) - 12*(n-2)*(1255*n^3 - 3957*n^2 + 4492*n - 1820)*a(n-2) + 2*(n-3)*(n-2)*(10655*n^2 - 32733*n + 26908)*a(n-3) - 4*(n-4)*(n-3)*(n-2)*(3445*n - 6986)*a(n-4) + 3381*(n-5)*(n-4)*(n-3)*(n-2)*a(n-5). - Vaclav Kotesovec, Nov 17 2021