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A057625
a(n) = n! * sum 1/k! where the sum is over all positive integers k that divide n.
31
1, 3, 7, 37, 121, 1201, 5041, 62161, 423361, 5473441, 39916801, 818959681, 6227020801, 130784734081, 1536517382401, 32256486662401, 355687428096001, 10679532671808001, 121645100408832001, 3770998783116364801, 59616236292028416001, 1686001119824999577601
OFFSET
1,2
COMMENTS
Sets of lists of equal size, cf. A000262. - Vladeta Jovovic, Nov 02 2003
From Gus Wiseman, Jan 10 2019: (Start)
Number of matrices whose entries are 1,...,n, up to column permutations. For example, inequivalent representatives of the a(4) = 37 matrices are:
One 1 X 4 matrix:
[1234]
12 2 X 2 matrices:
[12] [12] [13] [13] [14] [14] [23] [23] [24] [24] [34] [34]
[34] [43] [24] [42] [23] [32] [14] [41] [13] [31] [12] [21]
and 24 4 X 1 matrices:
[1][1][1][1][1][1][2][2][2][2][2][2][3][3][3][3][3][3][4][4][4][4][4][4]
[2][2][3][3][4][4][1][1][3][3][4][4][1][1][2][2][4][4][1][1][2][2][3][3]
[3][4][2][4][2][3][3][4][1][4][1][3][2][4][1][4][1][2][2][3][1][3][1][2]
[4][3][4][2][3][2][4][3][4][1][3][1][4][2][4][1][2][1][3][2][3][1][2][1]
in total 1+12+24 = 37.
(End)
LINKS
FORMULA
E.g.f.: Sum_{n>0} (exp(x^n)-1). - Vladeta Jovovic, Dec 30 2001
E.g.f.: Sum_{k>0} x^k/k!/(1-x^k). - Vladeta Jovovic, Oct 14 2003
Equals the logarithmic derivative of A209903. - Paul D. Hanna, Jul 26 2012
EXAMPLE
a(4) = 4! (1 + 1/2! + 1/4!) = 24 (1 + 1/2 + 1/24) = 37.
MATHEMATICA
a[n_] := n! DivisorSum[n, 1/#! &]; Array[a, 22] (* Jean-François Alcover, Dec 23 2015 *)
PROG
(PARI) a(n)=n! * sumdiv(n, d, 1/d! ); /* Joerg Arndt, Oct 07 2012 */
CROSSREFS
KEYWORD
nonn
AUTHOR
Leroy Quet, Oct 09 2000
STATUS
approved