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A132963 Total number of distinct block sizes in all partitions of [n].

%I A132963 #14 Jan 06 2022 17:44:40
%S A132963 1,2,8,25,102,439,2067,10406,56754,328257,2015818,13067366,89192170,
%T A132963 638321285,4779442602,37332643831,303635437532,2565592977205,
%U A132963 22483754207839,204013083946460,1913880812797792,18536832515581167,185130415180288134,1904280138346826637
%N A132963 Total number of distinct block sizes in all partitions of [n].
%H A132963 G. C. Greubel, <a href="/A132963/b132963.txt">Table of n, a(n) for n = 1..500</a>
%F A132963 E.g.f.: exp(exp(x)-1)*Sum_{k>0} (1-exp(-x^k/k!)).
%p A132963 b:= proc(n, i, c) option remember; `if`(n=0, c,
%p A132963       `if`(i<1, 0, add(b(n-j*i, i-1, c+signum(j))*
%p A132963       combinat[multinomial](n, n-i*j, i$j)/j!, j=0..n/i)))
%p A132963     end:
%p A132963 a:= n-> b(n$2, 0):
%p A132963 seq(a(n), n=1..30);  # _Alois P. Heinz_, Jan 06 2022
%t A132963 Rest[ Range[0, 23]! CoefficientList[ Series[ Exp[ Exp[x] - 1] Sum[1 - Exp[ -x^k/k! ], {k, 30}], {x, 0, 23}], x]] (* _Robert G. Wilson v_, Sep 13 2007 *)
%Y A132963 Cf. A005493, A132958, A132959, A132960, A132961, A132962, A350175.
%K A132963 easy,nonn
%O A132963 1,2
%A A132963 _Vladeta Jovovic_, Sep 06 2007
%E A132963 More terms from _Robert G. Wilson v_, Sep 13 2007