A381379 E.g.f. A(x) satisfies A(x) = 1/( 1 - x * cos(x * A(x)) )^2.

1, 2, 6, 18, -48, -2630, -52800, -824054, -8682240, 54462258, 7410631680, 305163480578, 8935815871488, 167137193150954, -1440976761090048, -349400091225243270, -22113174143289262080, -960586728800597050526, -26252145855684866211840, 255024367557922004307442

Offset

0,2

Comments

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

Links

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

Formula

E.g.f.: B(x)^2, where B(x) is the e.g.f. of A381378.

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

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, Oct 26 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) = 2*sum(k=0, n, k!*binomial(2*n-k+2, k)/(2*n-k+2)*I^(n-k)*a185951(n, k));

Crossrefs

Cf. A185951, A381378.

Sequence in context: A377118 A256828 A197055 * A258625 A062026 A048495

Adjacent sequences: A381376 A381377 A381378 * A381380 A381381 A381382

Keyword

sign

Author

Seiichi Manyama, Feb 22 2025

Status

approved