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%I #80 Mar 18 2022 13:05:52
%S 1,1,1,4,5,16,82,169,541,2272,17966,44419,201830,802751,4897453,
%T 52275409,166257661,840363296,4321172134,24358246735,183351656650,
%U 2762567051857,10112898715063,62269802986835,343651382271526,2352104168848091,15649414071734847
%N Number of partitions of n-set with distinct block sizes.
%C Conjecture: the Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for all primes p and positive integers n and k. Cf. A185895. - _Peter Bala_, Mar 17 2022
%H Alois P. Heinz, <a href="/A007837/b007837.txt">Table of n, a(n) for n = 0..700</a>
%H Philippe Flajolet, Éric Fusy, Xavier Gourdon, Daniel Panario and Nicolas Pouyanne, <a href="http://arxiv.org/abs/math/0606370">A Hybrid of Darboux's Method and Singularity Analysis in Combinatorial Asymptotics</a>, Fig. 3, arXiv:math/0606370 [math.CO], 2006.
%H Knopfmacher, A., Odlyzko, A. M., Pittel, B., Richmond, L. B., Stark, D., Szekeres, G. and Wormald, N. C., <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v6i1r2">The asymptotic number of set partitions with unequal block sizes</a>, Electron. J. Combin., 6 (1999), no. 1, Research Paper 2, 36 pp.
%H Gus Wiseman, <a href="/A038041/a038041.txt">Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.</a>
%F E.g.f.: Product_{m >= 1} (1+x^m/m!).
%F a(n) = Sum_{k=1..n} (n-1)!/(n-k)!*b(k)*a(n-k), where b(k) = Sum_{d divides k} (-d)*(-d!)^(-k/d) and a(0) = 1. - _Vladeta Jovovic_, Oct 13 2002
%F E.g.f.: exp(Sum_{k>=1} Sum_{j>=1} (-1)^(k+1)*x^(j*k)/(k*(j!)^k)). - _Ilya Gutkovskiy_, Jun 18 2018
%e From _Gus Wiseman_, Jul 13 2019: (Start)
%e The a(1) = 1 through a(5) = 16 set partitions with distinct block sizes:
%e {{1}} {{1,2}} {{1,2,3}} {{1,2,3,4}} {{1,2,3,4,5}}
%e {{1},{2,3}} {{1},{2,3,4}} {{1},{2,3,4,5}}
%e {{1,2},{3}} {{1,2,3},{4}} {{1,2},{3,4,5}}
%e {{1,3},{2}} {{1,2,4},{3}} {{1,2,3},{4,5}}
%e {{1,3,4},{2}} {{1,2,3,4},{5}}
%e {{1,2,3,5},{4}}
%e {{1,2,4},{3,5}}
%e {{1,2,4,5},{3}}
%e {{1,2,5},{3,4}}
%e {{1,3},{2,4,5}}
%e {{1,3,4},{2,5}}
%e {{1,3,4,5},{2}}
%e {{1,3,5},{2,4}}
%e {{1,4},{2,3,5}}
%e {{1,4,5},{2,3}}
%e {{1,5},{2,3,4}}
%e (End)
%p a:= proc(n) option remember; `if`(n=0, 1, add(add((-d)*(-d!)^(-k/d),
%p d=numtheory[divisors](k))*(n-1)!/(n-k)!*a(n-k), k=1..n))
%p end:
%p seq(a(n), n=0..30); # _Alois P. Heinz_, Sep 06 2008
%p # second Maple program:
%p A007837 := proc(n) option remember; local k; `if`(n = 0, 1,
%p add(binomial(n-1, k-1) * A182927(k) * A007837(n-k), k = 1..n)) end:
%p seq(A007837(i),i=0..24); # _Peter Luschny_, Apr 25 2011
%t nn=20;p=Product[1+x^i/i!,{i,1,nn}];Drop[Range[0,nn]!CoefficientList[ Series[p,{x,0,nn}],x],1] (* _Geoffrey Critzer_, Sep 22 2012 *)
%t a[0]=1; a[n_] := a[n] = Sum[(n-1)!/(n-k)!*DivisorSum[k, -#*(-#!)^(-k/#)&]* a[n-k], {k, 1, n}]; Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Nov 23 2015, after _Vladeta Jovovic_ *)
%o (PARI) {my(n=20); Vec(serlaplace(prod(k=1, n, (1+x^k/k!) + O(x*x^n))))} \\ _Andrew Howroyd_, Dec 21 2017
%Y Row sums of A131632 or A262072 or A262078 or A309992.
%Y Cf. A000110, A005651, A007838, A032011, A035470, A038041, A178682, A265950, A271423, A275780, A326026, A326514, A326517, A326533.
%Y Column k=0 of A327869.
%K nonn
%O 0,4
%A _Arnold Knopfmacher_
%E More terms from _Christian G. Bower_
%E a(0)=1 prepended by _Alois P. Heinz_, Aug 29 2015
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