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A130263 Number of degree-n permutations such that number of cycles of size k is odd (or zero) for every k. 10
1, 1, 1, 6, 14, 85, 529, 3451, 26816, 243909, 2507333, 26196841, 323194816, 4086482335, 57669014597, 864137455455, 13792308331616, 231648908415001, 4211676768746569, 79205041816808905, 1584565388341689032, 33265011234209710011, 730971789582886971689 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
LINKS
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
Sequence in context: A182752 A200033 A219376 * A213681 A308489 A301425
KEYWORD
easy,nonn
AUTHOR
Vladeta Jovovic, Aug 06 2007
EXTENSIONS
More terms from Emeric Deutsch, Aug 24 2007
STATUS
approved

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Last modified March 2 21:56 EST 2024. Contains 370498 sequences. (Running on oeis4.)