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 A120732 Number of square matrices with nonnegative integer entries and without zero rows or columns such that sum of all entries is equal to n. 27
 1, 1, 3, 15, 107, 991, 11267, 151721, 2360375, 41650861, 821881709, 17932031225, 428630422697, 11138928977049, 312680873171465, 9428701154866535, 303957777464447449, 10431949496859168189, 379755239311735494421 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 LINKS G. C. Greubel, Table of n, a(n) for n = 0..250 FORMULA a(n) = (1/n!)*Sum_{k=0..n} (-1)^(n-k)*Stirling1(n,k)*A048144(k). G.f.: Sum_{n>=0} Sum_{j=0..n} (-1)^(n-j)*binomial(n,j)*((1-x)^(-j)-1)^n. a(n) ~ c * n! / (sqrt(n) * (log(2))^(2*n)), where c = 0.4670932578797312973586879293426... . - Vaclav Kotesovec, May 07 2014 In closed form, c = 2^(log(2)/2-2) / (log(2) * sqrt(Pi*(1-log(2)))). - Vaclav Kotesovec, May 03 2015 G.f.: Sum_{n>=0} (1-x)^n * (1 - (1-x)^n)^n. - Paul D. Hanna, Mar 26 2018 EXAMPLE From Gus Wiseman, Nov 14 2018: (Start) The a(3) = 15 matrices:   [3] .   [2 0] [1 1] [1 1] [1 0] [1 0] [0 2] [0 1] [0 1]   [0 1] [1 0] [0 1] [1 1] [0 2] [1 0] [2 0] [1 1] .   [1 0 0] [1 0 0] [0 1 0] [0 1 0] [0 0 1] [0 0 1]   [0 1 0] [0 0 1] [1 0 0] [0 0 1] [1 0 0] [0 1 0]   [0 0 1] [0 1 0] [0 0 1] [1 0 0] [0 1 0] [1 0 0] (End) MATHEMATICA Table[1/n!*Sum[(-1)^(n-k)*StirlingS1[n, k]*Sum[(m!)^2*StirlingS2[k, m]^2, {m, 0, k}], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, May 07 2014 *) multsubs[set_, k_]:=If[k==0, {{}}, Join@@Table[Prepend[#, set[[i]]]&/@multsubs[Drop[set, i-1], k-1], {i, Length[set]}]]; Table[Length[Select[multsubs[Tuples[Range[n], 2], n], Union[First/@#]==Union[Last/@#]==Range[Max@@First/@#]&]], {n, 5}] (* Gus Wiseman, Nov 14 2018 *) CROSSREFS Cf. A007716, A048291, A054976, A057149, A057150, A057151, A104601, A104602, A120733, A138178, A316983, A319616. Sequence in context: A120016 A074519 A105618 * A245835 A090351 A136221 Adjacent sequences:  A120729 A120730 A120731 * A120733 A120734 A120735 KEYWORD nonn AUTHOR Vladeta Jovovic, Aug 18 2006 STATUS approved

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Last modified August 4 09:47 EDT 2021. Contains 346446 sequences. (Running on oeis4.)