A381378 E.g.f. A(x) satisfies A(x) = 1/( 1 - x * cos(x * A(x)^2) ).

1, 1, 2, 3, -48, -1135, -18240, -231637, -1356544, 53849889, 3026119680, 100808786419, 2429052865536, 26284690243825, -1539261873164288, -140633348417624805, -7196339681250508800, -258335768147494234303, -4225401456668904259584, 307227604973975435785571

Offset

0,3

Comments

As stated in the comment of A185951, A185951(n,0) = 0^n.

Links

Table of n, a(n) for n=0..19.

Formula

a(n) = Sum_{k=0..n} k! * binomial(2*n-k+1,k)/(2*n-k+1) * i^(n-k) * A185951(n,k), where i is the imaginary unit.

Maple

a:= n-> n!*coeff(series(RootOf(1/(1-x*cos(x*A^2))-A, A), x, n+1), x, n):
seq(a(n), n=0..19);  # Alois P. Heinz, Nov 24 2025

Mathematica

terms = 20; A[_] = 0; Do[A[x_] = 1/( 1 - x * Cos[x * A[x]^2]) + O[x]^terms // Normal, terms]; CoefficientList[A[x], x]*Range[0, terms-1]! (* Stefano Spezia, Nov 24 2025 *)

Prog

(PARI) a185951(n, k) = binomial(n, k)/2^k*sum(j=0, k, (2*j-k)^(n-k)*binomial(k, j));
a(n) = sum(k=0, n, k!*binomial(2*n-k+1, k)/(2*n-k+1)*I^(n-k)*a185951(n, k));

Crossrefs

Cf. A364980, A381376, A381382, A381384.

Cf. A185951.

Sequence in context: A173355 A118222 A381286 * A191594 A208204 A067092

Adjacent sequences: A381375 A381376 A381377 * A381379 A381380 A381381

Keyword

sign

Author

Seiichi Manyama, Feb 22 2025

Status

approved