A381182 -id:A381182 - cp's OEIS frontend

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

Seiichi Manyama, Feb 16 2025

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Index entries for reversions of series

Cf. A201627, A381180, A381182.
Cf. A162653, A381171, A381173.
Cf. A136630, A196776.

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);

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.

A381180 E.g.f. A(x) satisfies A(x) = 1 + sin(x*A(x)) / A(x).

Original entry on oeis.org

1, 1, 0, -1, -8, -19, 64, 1091, 7680, -1415, -650752, -8575865, -35559424, 857890021, 21380186112, 203548592651, -1615715926016, -95486152906639, -1599622990659584, -1397194164399601, 657963431581974528, 18168041375501245021, 157453907927886725120, -6059840564222790027821

Offset: 0

Seiichi Manyama, Feb 16 2025

sign

Cf. A201627, A381181, A381182.

Cf. A136630, A196776.

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-k+1, k)/(n-k+1)*I^(n-k)*a136630(n, k));

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

cp's OEIS Frontend

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

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