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A007841 Number of factorizations of permutations of n letters into cycles in nondecreasing length order.

%I A007841 #49 Jul 02 2025 16:01:55
%S A007841 1,1,3,11,56,324,2324,18332,167544,1674264,18615432,223686792,
%T A007841 2937715296,41233157952,623159583552,10008728738304,171213653641344,
%U A007841 3092653420877952,59086024678203264,1185657912197967744,25015435198774723584,552130504313534175744
%N A007841 Number of factorizations of permutations of n letters into cycles in nondecreasing length order.
%H A007841 Alois P. Heinz, <a href="/A007841/b007841.txt">Table of n, a(n) for n = 0..450</a>
%H A007841 Vaclav Kotesovec, <a href="/A007841/a007841.jpg">Graph - The asymptotic ratio</a>
%H A007841 A. Knopfmacher, J. N. Ridley, <a href="http://dx.doi.org/10.1137/0406031">Reciprocal sums over partitions and compositions</a>, SIAM J. Discrete Math. 6 (1993), no. 3, 388-399.
%H A007841 D. H. Lehmer, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa21/aa21123.pdf">On reciprocally weighted partitions</a>, Acta Arithmetica XXI (1972), 379-388.
%F A007841 E.g.f.: prod{m >= 1} 1/(1-x^m/m).
%F A007841 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
%F A007841 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
%F A007841 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
%F A007841 E.g.f.: exp(Sum_{k>=1} Sum_{j>=1} x^(j*k)/(k*j^k)). - _Ilya Gutkovskiy_, May 27 2018
%p A007841 p := product(1/(1-x^m/m), m=1..100):
%p A007841 s := series(p,x,100):
%p A007841 for i from 0 to 100 do printf(`%.0f,`,i!*coeff(s,x,i)) od:
%p A007841 # second Maple program:
%p A007841 with(combinat):
%p A007841 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
%p A007841      (i-1)!^j*b(n-i*j, i-1)*multinomial(n, n-i*j, i$j), j=0..n/i)))
%p A007841     end:
%p A007841 a:= n-> b(n$2):
%p A007841 seq(a(n), n=0..30);  # _Alois P. Heinz_, Jul 21 2014
%t A007841 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 *)
%t A007841 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 *)
%o A007841 (PARI)
%o A007841 N=66; q='q+O('q^N);
%o A007841 f=1/prod(n=1,N, 1-1/n*q^n );
%o A007841 egf=serlaplace(f);
%o A007841 Vec(egf)
%o A007841 /* _Joerg Arndt_, Oct 06 2012 */
%o A007841 (Maxima)
%o A007841 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);
%o A007841 makelist(R(n,1),n,0,21); /* _Vladimir Kruchinin_, Sep 09 2014 */
%Y A007841 Cf. A007837, A007838, A249078, A249480, A249588, A249593, A269791, A269793, A269794.
%K A007841 nonn
%O A007841 0,3
%A A007841 _Arnold Knopfmacher_
%E A007841 More terms from _James Sellers_, Jan 09 2001
%E A007841 Prepended a(0) = 1, _Joerg Arndt_, Oct 06 2012