A235128 Expansion of e.g.f. 1/(1 - sin(7*x))^(1/7).

1, 1, 8, 71, 1072, 20161, 476288, 13315751, 432387712, 15959926081, 660372282368, 30265936565831, 1522069164439552, 83327826089289601, 4933286107483701248, 314052936209639958311, 21392225375507849838592, 1552501782546292090638721, 119588747474281844162428928

Offset: 0

Author

Vaclav Kotesovec, Jan 03 2014

Keywords

• nonn
• easy

Comments

Generally, for e.g.f. 1/(1-sin(p*x))^(1/p) we have a(n) ~ n! * 2^(n+3/p) * p^n / (Gamma(2/p) * n^(1-2/p) * Pi^(n+2/p)).

Crossrefs

Cf. A001586 (p=2), A007788 (p=3), A144015 (p=4), A230134 (p=5), A227544 (p=6), A230114 (p=8).

Cf. A045754, A136630.

Programs

Mathematica

CoefficientList[Series[1/(1-Sin[7*x])^(1/7), {x, 0, 20}], x] * Range[0, 20]!

PARI

a136630(n, k) = 1/(2^k*k!)*sum(j=0, k, (-1)^(k-j)*(2*j-k)^n*binomial(k, j));
a045754(n) = prod(k=0, n-1, 7*k+1);
a(n) = sum(k=0, n, a045754(k)*(7*I)^(n-k)*a136630(n, k)); \\ Seiichi Manyama, Jun 24 2025

Formula

a(n) ~ n! * 2^(n+3/7) * 7^n / (Gamma(2/7) * n^(5/7) * Pi^(n+2/7)).

a(n) = Sum_{k=0..n} A045754(k) * (7*i)^(n-k) * A136630(n,k), where i is the imaginary unit. - Seiichi Manyama, Jun 24 2025