A355775 Expansion of 1 + f/(1 + 2*f), where f is the g.f. of the swinging factorial (A056040).

1, 1, 0, 2, -10, 30, -84, 244, -722, 2126, -6252, 18364, -53988, 158668, -466408, 1370792, -4029154, 11842222, -34806972, 102303468, -300690348, 883782404, -2597604952, 7634834648, -22440207764, 65955900268, -193856647736, 569780485080, -1674690451976

Offset: 0

Author

Peter Luschny, Jul 22 2022

Keywords

• sign

Crossrefs

Cf. A056040, A355488.

Programs

Maple

# Traditional style:
u := 1 - 4*z^2: v := u + z: w := u^(3/2) - v: 1 + w/(w - v):
ser := series(%, z, 30): seq(coeff(%, z, n), n = 0..28);
# Alternative (Alois P. Heinz style, see A355488):
c := n -> n! / iquo(n, 2)!^2:
a := n -> (f -> coeff(series(1 + f/(1 + 2*f), x, n+1), x, n))(add(c(j)*x^j, j=1..n)): seq(a(n), n = 0..28);

Mathematica

c[n_] := n! / Quotient[n, 2]!^2;
a[n_] := With[{f = Sum[c[j]*x^j, {j, 1, n}]}, SeriesCoefficient[1 + f/(1 + 2*f), {x, 0, n}]];
Table[a[n], {n, 0, 28}] (* Jean-François Alcover, Apr 17 2025, after Alois P. Heinz *)

SageMath

def sw_factorial(n): return factorial(n) // factorial(n // 2)^2
A = QQ[['t']]
f = A([0] + [sw_factorial(n) for n in range(1, 29)]).O(29)
print(list(1 + f / (1 + 2 * f)))  # After F. Chapoton in A355488.

Formula

a(n) = [z^n] (1 + ((1 - 4*z^2)^(3/2) + 4*z^2 - z - 1) / ((1 - 4*z^2)^(3/2) + 2*(4*z^2 - z - 1))).

Let u = 1 - 4*z^2, v = u + z and w = u^(3/2) - v. Then the g.f. can be written as 1 + w/(w - v) and the g.f. of the swinging factorial 1 - w/(w + v).