A385305 Expansion of e.g.f. 1/(1 - 3 * sinh(x))^(1/3).

1, 1, 4, 29, 296, 3921, 63904, 1236509, 27700096, 705098241, 20100847104, 634406699389, 21959759364096, 827184049670161, 33684401687855104, 1474548883501060669, 69051807696652599296, 3444499079760040247681, 182339939994632235515904, 10209271857672376613472349

Offset: 0

Author

Seiichi Manyama, Jun 24 2025

Keywords

• nonn

Crossrefs

Cf. A006154, A385304.

Cf. A380017, A385307, A385309, A385311.

Cf. A007559, A136630, A235135.

Programs

PARI

a136630(n, k) = 1/(2^k*k!)*sum(j=0, k, (-1)^(k-j)*(2*j-k)^n*binomial(k, j));
a007559(n) = prod(k=0, n-1, 3*k+1);
a(n) = sum(k=0, n, a007559(k)*a136630(n, k));

Formula

a(n) = Sum_{k=0..n} A007559(k) * A136630(n,k).

a(n) ~ sqrt(Pi) * 2^(1/3) * n^(n - 1/6) / (5^(1/6) * Gamma(1/3) * exp(n) * log((1 + sqrt(10))/3)^(n + 1/3)). - Vaclav Kotesovec, Jun 28 2025