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%I #18 Sep 14 2019 16:48:46
%S 1,1,4,12,72,240,2880,10080,161280,1088640,14515200,79833600,
%T 2874009600,12454041600,348713164800,5230697472000,104613949440000,
%U 711374856192000,38414242234368000,243290200817664000,14597412049059840000,204363768686837760000
%N Number of ways to fill a matrix with the first n positive integers.
%F a(n) = A000005(n) * n! for n > 0, a(0) = 1.
%F E.g.f.: 1 + Sum_{k>=1} x^k/(1 - x^k). - _Ilya Gutkovskiy_, Sep 13 2019
%e The a(4) = 72 matrices consist of:
%e 24 row/column permutations of [1 2 3 4]
%e +
%e 4 row/column permutations of [1 2]
%e [3 4]
%e +
%e 4 row/column permutations of [1 2]
%e [4 3]
%e +
%e 4 row/column permutations of [1 3]
%e [2 4]
%e +
%e 4 row/column permutations of [1 3]
%e [4 2]
%e +
%e 4 row/column permutations of [1 4]
%e [2 3]
%e +
%e 4 row/column permutations of [1 4]
%e [3 2]
%e +
%e 24 row/column permutations of [1]
%e [2]
%e [3]
%e [4]
%t Join[{1}, Table[DivisorSigma[0, n]*n!, {n, 30}]]
%o (PARI) a(n) = if (n==0, 1, numdiv(n)*n!); \\ _Michel Marcus_, Jan 15 2019
%Y Cf. A000005, A000142, A038041, A053529, A057625, A061095, A120733, A121860.
%Y Cf. A323300, A323301, A323307, A323351.
%K nonn
%O 0,3
%A _Gus Wiseman_, Jan 12 2019