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%I #9 Jul 22 2022 10:38:55
%S 1,1,0,2,-10,30,-84,244,-722,2126,-6252,18364,-53988,158668,-466408,
%T 1370792,-4029154,11842222,-34806972,102303468,-300690348,883782404,
%U -2597604952,7634834648,-22440207764,65955900268,-193856647736,569780485080,-1674690451976
%N Expansion of 1 + f/(1 + 2*f), where f is the g.f. of the swinging factorial (A056040).
%F 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))).
%F 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).
%p # Traditional style:
%p u := 1 - 4*z^2: v := u + z: w := u^(3/2) - v: 1 + w/(w - v):
%p ser := series(%, z, 30): seq(coeff(%, z, n), n = 0..28);
%p # Alternative (_Alois P. Heinz_ style, see A355488):
%p c := n -> n! / iquo(n, 2)!^2:
%p 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);
%o (SageMath)
%o def sw_factorial(n): return factorial(n) // factorial(n // 2)^2
%o A = QQ[['t']]
%o f = A([0] + [sw_factorial(n) for n in range(1, 29)]).O(29)
%o print(list(1 + f / (1 + 2 * f))) # After _F. Chapoton_ in A355488.
%Y Cf. A056040, A355488.
%K sign
%O 0,4
%A _Peter Luschny_, Jul 22 2022
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