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A286071 Number of permutations of [n] with nonincreasing cycle sizes.

%I A286071 #11 May 24 2018 02:56:25
%S A286071 1,1,2,5,19,85,496,3229,25117,215225,2100430,22187281,261228199,
%T A286071 3284651245,45163266604,659277401525,10380194835601,172251467909809,
%U A286071 3057368096689690,56867779157145769,1122474190194034555,23137433884903034501,502874858021076645784
%N A286071 Number of permutations of [n] with nonincreasing cycle sizes.
%C A286071 Each cycle is written with the smallest element first and cycles are arranged in increasing order of their first elements.
%C A286071 a(n) is even if and only if n in { A016825 }.
%H A286071 Alois P. Heinz, <a href="/A286071/b286071.txt">Table of n, a(n) for n = 0..450</a>
%H A286071 Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation">Permutation</a>
%e A286071 a(3) = 5: (123), (132), (12)(3), (13)(2), (1)(2)(3).
%e A286071 a(4) = 19: (1234), (1243), (1324), (1342), (1423), (1432), (123)(4), (132)(4), (124)(3), (142)(3), (12)(34), (12)(3)(4), (134)(2), (143)(2), (13)(24), (13)(2)(4), (14)(23), (14)(2)(3), (1)(2)(3)(4).
%p A286071 b:= proc(n, i) option remember; `if`(n=0, 1, add((j-1)!*
%p A286071       b(n-j, j)*binomial(n-1, j-1), j=1..min(n, i)))
%p A286071     end:
%p A286071 a:= n-> b(n$2):
%p A286071 seq(a(n), n=0..30);
%t A286071 b[n_, i_] := b[n, i] = If[n == 0, 1, Sum[(j - 1)!*b[n - j, j]*Binomial[n - 1, j - 1], {j, 1, Min[n, i]}]];
%t A286071 a[n_] := b[n, n];
%t A286071 Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, May 24 2018, translated from Maple *)
%Y A286071 Cf. A016825, A275310, A286072, A286073, A286074, A286075, A286076, A286077.
%K A286071 nonn
%O A286071 0,3
%A A286071 _Alois P. Heinz_, May 01 2017