A349103 Expansion of e.g.f. exp( (sin(x) + x*cos(x))/2 ).

1, 1, 1, -1, -7, -16, 19, 270, 809, -1832, -26939, -81680, 415817, 5085192, 11943199, -154023064, -1510744111, -1192784896, 80798373289, 615743987520, -1348945040647, -55570193758592, -296824726927253, 2567750854163200, 47111874571146041

Offset: 0

Author

Seiichi Manyama, Mar 26 2022

Keywords

• sign

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..571

Crossrefs

Cf. A002017, A009189, A349104, A351937.

Programs

Mathematica

With[{m = 24}, Range[0, m]! * CoefficientList[Series[Exp[(Sin[x] + x*Cos[x])/2], {x, 0, m}], x]] (* Amiram Eldar, Mar 26 2022 *)

PARI

my(N=40, x='x+O('x^N)); Vec(serlaplace(exp((sin(x)+x*cos(x))/2)))

PARI

a(n) = if(n==0, 1, sum(k=0, (n-1)\2, (-1)^k*(k+1)*binomial(n-1, 2*k)*a(n-2*k-1)));

Formula

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