A381181 Expansion of e.g.f. (1/x) * Series_Reversion( x / (1 + sin(x)) ).

1, 1, 2, 5, 8, -79, -1584, -20539, -223616, -1855295, -1736960, 435730789, 14511117312, 338965239601, 6202042886144, 71638247035109, -714560796196864, -84697775518956799, -3650903032332091392, -115829159202293866939, -2739961030150105333760, -29414406825401517785039

Offset: 0

Author

Seiichi Manyama, Feb 16 2025

Keywords

• sign

Links

• Index entries for reversions of series

Crossrefs

Cf. A201627, A381180, A381182.

Cf. A162653, A381171, A381173.

Cf. A136630, A196776.

Programs

PARI

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

Formula

E.g.f. A(x) satisfies A(x) = 1 + sin(x * A(x)).

a(n) = (1/(n+1)) * Sum_{k=0..n} k! * binomial(n+1,k) * i^(n-k) * A136630(n,k), where i is the imaginary unit.