A059849 -id:A059849 - cp's OEIS frontend

A318395 Number of nonnegative integer matrices with values summing to n, up to transposition and permutation of rows and columns.

1, 1, 3, 7, 21, 54, 167, 491, 1586, 5132, 17442, 60399, 216172, 790436, 2965333, 11365813, 44536775, 178107679, 726716229, 3022464373, 12807206008, 55253891494, 242585471236, 1083255591604, 4917631017573, 22685090928596, 106291554085987, 505653658171936, 2441383079595849

Offset: 0

Gus Wiseman, Aug 25 2018

nonn

Also the number of non-isomorphic pairs of set partitions of {1,...,n}.

			Inequivalent representatives of the a(3) = 7 nonnegative integer matrices:
  [3]   [1 2]   [1 1 1]   [1 0]   [0 1]   [1 0 0]   [1 0 0]
                          [0 2]   [1 1]   [0 1 1]   [0 1 0]
                                                    [0 0 1]
Non-isomorphic representatives of the a(3) = 7 pairs of set partitions:
    {{1,2,3}}     {{1,2,3}}
    {{1,2,3}}    {{1},{2,3}}
    {{1,2,3}}   {{1},{2},{3}}
   {{1},{2,3}}   {{1},{2,3}}
   {{1},{2,3}}   {{2},{1,3}}
   {{1},{2,3}}  {{1},{2},{3}}
  {{1},{2},{3}} {{1},{2},{3}}

Andrew Howroyd, Table of n, a(n) for n = 0..50

Cf. A000110, A000258, A001247, A007716, A008277, A049311, A059849, A060639, A116540, A181939, A316983, A318393.

a(n) = (A007716(n) + A316983(n))/2. - Andrew Howroyd, Sep 03 2018

a(6)-a(25) from Andrew Howroyd, Sep 03 2018

Terms a(26) and beyond from Andrew Howroyd, Mar 29 2020

A122486 a(n) = Sum_{k=0..n} |Stirling1(n,k)|*Bell(k)^2.

Original entry on oeis.org

1, 1, 5, 39, 425, 6053, 107735, 2321469, 59152987, 1750362419, 59286010621, 2271617296347, 97502863649141, 4649359584613201, 244550369307356039, 14101227268075911837, 886551391533830227267, 60482082002935189216499

Offset: 0

Vladeta Jovovic, Sep 15 2006, Sep 19 2006

Row sums of the absolute values of the triangle of Stirling1(n,k)*Bell(k)^2:
  1;
  0,    1;
  0,   -1,     4;
  0,    2,   -12,    25;
  0,   -6,    44,  -150,     225;
  0,   24,  -200,   875,   -2250,   2704;
  0, -120,  1096, -5625,   19125, -40560,   41209;
  0,  720, -7056, 40600, -165375, 473200, -865389, 769129;
  ... - R. J. Mathar, Jan 27 2017

Cf. A000110, A059849.

with(combinat): seq(sum(abs(stirling1(n,k))*bell(k)^2,k=0..n),n=0..19); # Emeric Deutsch, Oct 08 2006

a(n) = exp(-2)*Sum_{r,s>=0} [r*s]^n/(r!*s!), where [m]^n = m*(m+1)*...*(m+n-1) is the rising factorial.

E.g.f.: Sum_{n>=0} exp( 1/(1-x)^n - 2 ) / n!. - Paul D. Hanna, Jul 25 2018

More terms from Emeric Deutsch, Oct 08 2006

cp's OEIS Frontend

A318395 Number of nonnegative integer matrices with values summing to n, up to transposition and permutation of rows and columns.

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