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A132961 Total number of all distinct cycle sizes in all permutations of [n].

%I A132961 #15 Aug 10 2025 00:56:46
%S A132961 1,2,9,38,215,1384,10409,86946,825075,8541998,97590779,1205343952,
%T A132961 16148472977,231416203212,3560209750005,58104163643054,
%U A132961 1008693571819919,18477578835352366,357476371577422955,7258865626801695048,154893910336866444009,3454112338490001478772
%N A132961 Total number of all distinct cycle sizes in all permutations of [n].
%H A132961 Alois P. Heinz, <a href="/A132961/b132961.txt">Table of n, a(n) for n = 1..450</a>
%F A132961 E.g.f.: 1/(1-x)*Sum_{k>0} (1-exp(-x^k/k)). Exponential convolution of A132960(n) and n!: a(n) = n!*Sum_{k=1..n} A132960(k)/k!.
%p A132961 with(combinat):
%p A132961 b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, 0,
%p A132961       add(multinomial(n, n-i*j, i$j)/j!*(i-1)!^j*(p-> p+
%p A132961       [0, p[1]*`if`(j>0, 1, 0)])(b(n-i*j, i-1)), j=0..n/i)))
%p A132961     end:
%p A132961 a:= n-> b(n$2)[2]:
%p A132961 seq(a(n), n=1..30);  # _Alois P. Heinz_, Oct 21 2015
%t A132961 Rest[ Range[0, 22]! CoefficientList[ Series[1/(1 - x) Sum[1 - Exp[ -x^k/k], {k, 25}], {x, 0, 22}], x]] (* _Robert G. Wilson v_, Sep 13 2007 *)
%Y A132961 Cf. A000254, A132958, A132959, A132960, A132962, A132963.
%Y A132961 Row sums of A293211.
%K A132961 nonn
%O A132961 1,2
%A A132961 _Vladeta Jovovic_, Sep 06 2007
%E A132961 More terms from _Robert G. Wilson v_, Sep 13 2007