A381180 -id:A381180 - 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

sign

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.

A381182 E.g.f. A(x) satisfies A(x) = 1/( 1 - A(x) * sin(x * A(x)) ).

Original entry on oeis.org

1, 1, 6, 71, 1288, 31661, 984640, 37085075, 1641305472, 83497838425, 4801347029504, 307975150996831, 21802395720298496, 1688562016007776261, 142023935786330431488, 12892154760586821775019, 1256251152910271399624704, 130793914073764385411654321, 14490427167940362294881615872

Offset: 0

Seiichi Manyama, Feb 16 2025

nonn

Cf. A201627, A381180, A381181.

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+2*k+1, k)/(n+2*k+1)*I^(n-k)*a136630(n, k));

a(n) = Sum_{k=0..n} k! * binomial(n+2*k+1,k)/(n+2*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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