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A057625 a(n) = n! * sum 1/k! where the sum is over all positive integers k that divide n.

%I A057625 #37 Feb 05 2019 16:26:53
%S A057625 1,3,7,37,121,1201,5041,62161,423361,5473441,39916801,818959681,
%T A057625 6227020801,130784734081,1536517382401,32256486662401,355687428096001,
%U A057625 10679532671808001,121645100408832001,3770998783116364801,59616236292028416001,1686001119824999577601
%N A057625 a(n) = n! * sum 1/k! where the sum is over all positive integers k that divide n.
%C A057625 Sets of lists of equal size, cf. A000262. - _Vladeta Jovovic_, Nov 02 2003
%C A057625 From _Gus Wiseman_, Jan 10 2019: (Start)
%C A057625 Number of matrices whose entries are 1,...,n, up to column permutations. For example, inequivalent representatives of the a(4) = 37 matrices are:
%C A057625 One 1 X 4 matrix:
%C A057625   [1234]
%C A057625 12 2 X 2 matrices:
%C A057625   [12] [12] [13] [13] [14] [14] [23] [23] [24] [24] [34] [34]
%C A057625   [34] [43] [24] [42] [23] [32] [14] [41] [13] [31] [12] [21]
%C A057625 and 24 4 X 1 matrices:
%C A057625   [1][1][1][1][1][1][2][2][2][2][2][2][3][3][3][3][3][3][4][4][4][4][4][4]
%C A057625   [2][2][3][3][4][4][1][1][3][3][4][4][1][1][2][2][4][4][1][1][2][2][3][3]
%C A057625   [3][4][2][4][2][3][3][4][1][4][1][3][2][4][1][4][1][2][2][3][1][3][1][2]
%C A057625   [4][3][4][2][3][2][4][3][4][1][3][1][4][2][4][1][2][1][3][2][3][1][2][1]
%C A057625 in total 1+12+24 = 37.
%C A057625 (End)
%H A057625 Seiichi Manyama, <a href="/A057625/b057625.txt">Table of n, a(n) for n = 1..449</a>
%F A057625 E.g.f.: Sum_{n>0} (exp(x^n)-1). - _Vladeta Jovovic_, Dec 30 2001
%F A057625 E.g.f.: Sum_{k>0} x^k/k!/(1-x^k). - _Vladeta Jovovic_, Oct 14 2003
%F A057625 Equals the logarithmic derivative of A209903. - _Paul D. Hanna_, Jul 26 2012
%e A057625 a(4) = 4! (1 + 1/2! + 1/4!) = 24 (1 + 1/2 + 1/24) = 37.
%t A057625 a[n_] := n! DivisorSum[n, 1/#! &]; Array[a, 22] (* _Jean-François Alcover_, Dec 23 2015 *)
%o A057625 (PARI) a(n)=n! * sumdiv(n, d, 1/d! );  /* _Joerg Arndt_, Oct 07 2012 */
%Y A057625 Cf. A000005, A005225, A038041, A038048, A061095, A121860, A209903 (exp), A236696, A320444.
%Y A057625 Row sums of A066387.
%K A057625 nonn
%O A057625 1,2
%A A057625 _Leroy Quet_, Oct 09 2000