A385307 Expansion of e.g.f. 1/(1 - 3 * sin(x))^(1/3).

1, 1, 4, 27, 264, 3361, 52704, 981707, 21176704, 519150241, 14255163904, 433384277787, 14451212550144, 524406240059521, 20572970822959104, 867641565719168267, 39145118179183427584, 1881294510800399083201, 95950279080398196834304, 5176039012712211526485147

Offset: 0

Author

Seiichi Manyama, Jun 24 2025

Keywords

• nonn

Crossrefs

Cf. A000111, A385306.

Cf. A380017, A385305, A385309, A385311.

Cf. A007559, A007788, A136630.

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)*I^(n-k)*a136630(n, k));

Formula

a(n) = Sum_{k=0..n} A007559(k) * i^(n-k) * A136630(n,k), where i is the imaginary unit.

a(n) ~ n! / (sqrt(2) * Gamma(1/3) * n^(2/3) * arcsin(1/3)^(n + 1/3)). - Vaclav Kotesovec, Jun 28 2025