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URL: https://oeis.org/A007841

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A007841
Number of factorizations of permutations of n letters into cycles in nondecreasing length order.
45
1, 1, 3, 11, 56, 324, 2324, 18332, 167544, 1674264, 18615432, 223686792, 2937715296, 41233157952, 623159583552, 10008728738304, 171213653641344, 3092653420877952, 59086024678203264, 1185657912197967744, 25015435198774723584, 552130504313534175744
OFFSET
0,3
LINKS
A. Knopfmacher, J. N. Ridley, Reciprocal sums over partitions and compositions, SIAM J. Discrete Math. 6 (1993), no. 3, 388-399.
D. H. Lehmer, On reciprocally weighted partitions, Acta Arithmetica XXI (1972), 379-388.
FORMULA
E.g.f.: prod{m >= 1} 1/(1-x^m/m).
a(n) = Sum_{k=1..n} (n-1)!/(n-k)!*b(k)*a(n-k), where b(k) = Sum_{d divides k} d^(1-k/d) and a(0) = 1. - Vladeta Jovovic, Oct 14 2002
a(n) = R(n,1), R(n,m) = R(n,m+1)+binomial(n,m)*(m-1)!*R(n-m,m), R(n,n)=(n-1)!, R(n,m)=0 for n<m. - Vladimir Kruchinin, Sep 09 2014
a(n) ~ c * n! * n, where c = exp(-gamma) = 0.56145948..., where gamma is the Euler-Mascheroni constant A001620 [Lehmer, 1972]. - Vaclav Kotesovec, Mar 05 2016
E.g.f.: exp(Sum_{k>=1} Sum_{j>=1} x^(j*k)/(k*j^k)). - Ilya Gutkovskiy, May 27 2018
MAPLE
p := product(1/(1-x^m/m), m=1..100):
s := series(p, x, 100):
for i from 0 to 100 do printf(`%.0f, `, i!*coeff(s, x, i)) od:
# Alternative:
with(combinat):
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
(i-1)!^j*b(n-i*j, i-1)*multinomial(n, n-i*j, i$j), j=0..n/i)))
end:
a:= n-> b(n$2):
seq(a(n), n=0..30); # Alois P. Heinz, Jul 21 2014
MATHEMATICA
nmax = 25; CoefficientList[Series[1/Product[(1 - x^k/k), {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Jul 24 2019 *)
nmax = 25; CoefficientList[Series[Exp[Sum[PolyLog[j, x^j]/j, {j, 1, nmax}]], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Jul 24 2019 *)
PROG
(PARI)
N=66; q='q+O('q^N);
f=1/prod(n=1, N, 1-1/n*q^n );
egf=serlaplace(f);
Vec(egf)
/* Joerg Arndt, Oct 06 2012 */
(Maxima)
R(n, m):=if n=0 then 1 else if n<m then 0 else if n=m then (n-1)! else R(n, m+1)+binomial(n, m)*(m-1)!*R(n-m, m);
makelist(R(n, 1), n, 0, 21); /* Vladimir Kruchinin, Sep 09 2014 */
KEYWORD
nonn
EXTENSIONS
More terms from James Sellers, Jan 09 2001
Prepended a(0) = 1, Joerg Arndt, Oct 06 2012
STATUS
approved