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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 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:=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. = 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

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