A130263 Number of degree-n permutations such that number of cycles of size k is odd (or zero) for every k.

1, 1, 1, 6, 14, 85, 529, 3451, 26816, 243909, 2507333, 26196841, 323194816, 4086482335, 57669014597, 864137455455, 13792308331616, 231648908415001, 4211676768746569, 79205041816808905, 1584565388341689032, 33265011234209710011, 730971789582886971689

Offset

0,4

Links

Alois P. Heinz, Table of n, a(n) for n = 0..450

Formula

E.g.f.: Product_{k>0} (1+sinh(x^k/k)).

a(n) ~ c * n!, where c = A270614 = Product_{k>=1} ((1 + sinh(1/k)) / exp(1/k)) = 0.625635801977949844... . - Vaclav Kotesovec, Mar 20 2016

Example

a(2)=1 because we have (12) ((1)(2) does not qualify). a(4)=14 because the following 10 permutations of 4 do not qualify: (1)(2)(3)(4), (14)(2)(3), (1)(24)(3), (1)(2)(34), (13)(2)(4), (13)(24), (1)(23)(4), (14)(23), (12)(3)(4) and (12)(34).

Maple

g:=product(1+sinh(x^k/k), k=1..40): gser:=series(g, x=0, 25): seq(factorial(n)*coeff(gser, x, n), n=0..21); # Emeric Deutsch, Aug 24 2007
# second Maple program:
with(combinat):
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(`if`(j=0 or irem(j, 2)=1, multinomial(n, n-i*j, i$j)
       *(i-1)!^j/j!*b(n-i*j, i-1), 0), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..30);  # Alois P. Heinz, Mar 09 2015

Mathematica

nn = 25; Range[0, nn]!*CoefficientList[Series[Product[1 + Sinh[x^k/k], {k, nn}], {x, 0, nn}], x] (* Vaclav Kotesovec, Mar 20 2016 *)

Prog

(Magma)
m:=40;
f:= func< x | (&*[1 + Sinh(x^j/j): j in [1..m+1]]) >;
R<x>:=PowerSeriesRing(Rationals(), m);
Coefficients(R!(Laplace( f(x) ))); // G. C. Greubel, Mar 18 2023
(SageMath)
m=40
def f(x): return product( 1 + sinh(x^j/j) for j in range(1, m+2) )
def A130263_list(prec):
    P.<x> = PowerSeriesRing(QQ, prec)
    return P( f(x) ).egf_to_ogf().list()
A130263_list(m) # G. C. Greubel, Mar 18 2023

Crossrefs

Cf. A055922, A130219, A130220, A130268, A218504, A270597, A270598, A270614.

Sequence in context: A182752 A200033 A219376 * A213681 A308489 A301425

Adjacent sequences: A130260 A130261 A130262 * A130264 A130265 A130266

Keyword

easy,nonn

Author

Vladeta Jovovic, Aug 06 2007

Extensions

More terms from Emeric Deutsch, Aug 24 2007

Status

approved