A354242 Expansion of e.g.f. 1/sqrt(5 - 4 * exp(x)).

1, 2, 14, 158, 2486, 50222, 1239254, 36126638, 1214933846, 46299580142, 1971815255894, 92809525295918, 4784166929982806, 268050260650705262, 16219498558371118934, 1054102762745609325998, 73229184033780135425366, 5415407651703010175897582

Offset: 0

Author

Seiichi Manyama, May 20 2022

Keywords

• nonn

Comments

From Peter Bala, Jul 07 2022: (Start)

Conjecture: Let k be a positive integer. The sequence obtained by reducing a(n) modulo k is eventually periodic with the period dividing phi(k) = A000010(k). For example, modulo 16 we obtain the sequence [1, 2, 14, 14, 6, 14, 6, 14, 6, ...], with an apparent period of 2 beginning at a(3). Cf. A354253.

More generally, we conjecture that the same property holds for integer sequences having an e.g.f. of the form G(exp(x) - 1), where G(x) is an integral power series. (End)

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..360

Crossrefs

Cf. A305404, A354252, A354253.

Cf. A000670, A001813, A094417, A316747, A354240, A354241.

Programs

PARI

my(N=20, x='x+O('x^N)); Vec(serlaplace(1/sqrt(5-4*exp(x))))

PARI

my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, binomial(2*k, k)*(exp(x)-1)^k)))

PARI

a(n) = sum(k=0, n, (2*k)!*stirling(n, k, 2)/k!);

Formula

E.g.f.: Sum_{k>=0} binomial(2*k,k) * (exp(x) - 1)^k.

a(n) = Sum_{k=0..n} (2*k)! * Stirling2(n,k)/k!.

a(n) ~ sqrt(2/5) * n^n / (exp(n) * log(5/4)^(n + 1/2)). - Vaclav Kotesovec, Jun 04 2022

Conjectural o.g.f. as a continued fraction of Stieltjes type: 1/(1 - 2*x/(1 - 5*x/(1 - 6*x/(1 - 10*x/(1 - 10*x/(1 - 15*x/(1 - ... - (4*n-2)*x/(1 - 5*n*x/(1 - ...))))))))). - Peter Bala, Jul 07 2022

a(0) = 1; a(n) = Sum_{k=1..n} (4 - 2*k/n) * binomial(n,k) * a(n-k). - Seiichi Manyama, Sep 09 2023

a(0) = 1; a(n) = 2*a(n-1) - 5*Sum_{k=1..n-1} (-1)^k * binomial(n-1,k) * a(n-k). - Seiichi Manyama, Nov 16 2023