A357738 Expansion of e.g.f. sin( 2 * (exp(x) - 1) )/2.

0, 1, 1, -3, -23, -83, -119, 973, 11145, 69805, 278281, 33165, -12794231, -157150355, -1271714807, -7108146611, -11364216951, 380051588653, 6923479542025, 78935931180813, 669998027706505, 3602978599128301, -8825050911646199, -598024924863875123

Offset

0,4

Links

Table of n, a(n) for n=0..23.

Eric Weisstein's MathWorld, Bell Polynomial.

Formula

a(n) = Sum_{k=0..floor((n-1)/2)} (-4)^(k) * Stirling2(n,2*k+1).

a(n) = 0; a(n) = Sum_{k=0..n-1} binomial(n-1, k) * A357727(k).

a(n) = ( Bell_n(2 * i) - Bell_n(-2 * i) )/(4 * i), where Bell_n(x) is n-th Bell polynomial and i is the imaginary unit.

Mathematica

With[{nn=30}, CoefficientList[Series[Sin[2(Exp[x]-1)]/2, {x, 0, nn}], x] Range[0, nn]!] (* Harvey P. Dale, Feb 19 2023 *)

Prog

(PARI) my(N=30, x='x+O('x^N)); concat(0, Vec(serlaplace(sin(2*(exp(x)-1))/2)))
(PARI) a(n) = sum(k=0, (n-1)\2, (-4)^k*stirling(n, 2*k+1, 2));
(PARI) Bell_poly(n, x) = exp(-x)*suminf(k=0, k^n*x^k/k!);
a(n) = round((Bell_poly(n, 2*I)-Bell_poly(n, -2*I)))/(4*I);

Crossrefs

Cf. A357598, A357727.

Sequence in context: A256329 A196649 A363170 * A027701 A201482 A032017

Adjacent sequences: A357735 A357736 A357737 * A357739 A357740 A357741

Keyword

sign

Author

Seiichi Manyama, Oct 11 2022

Status

approved