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A122400 Number of square (0,1)-matrices without zero rows and with exactly n entries equal to 1.

%I A122400 #37 Aug 09 2018 09:45:36
%S A122400 1,1,4,31,338,4769,82467,1687989,39905269,1069863695,32071995198,
%T A122400 1062991989013,38596477083550,1523554760656205,64961391010251904,
%U A122400 2975343608212835855,145687881987604377815,7594435556630244257213
%N A122400 Number of square (0,1)-matrices without zero rows and with exactly n entries equal to 1.
%H A122400 G. C. Greubel, <a href="/A122400/b122400.txt">Table of n, a(n) for n = 0..375</a>
%F A122400 a(n) = (1/n!)* Sum_{k=0..n} Stirling1(n,k)*A122399(k).
%F A122400 G.f.: Sum_{n>=0} ((1+x)^n - 1)^n. - _Vladeta Jovovic_, Sep 03 2006
%F A122400 G.f.: Sum_{n>=0} (1+x)^(n^2) / (1 + (1+x)^n)^(n+1). - _Paul D. Hanna_, Mar 23 2018
%F A122400 a(n) ~ c * d^n * n! / sqrt(n), where d = A317855 = (1+exp(1/r))*r^2 = 3.161088653865428813830172202588132491726382774188556341627278..., r = 0.8737024332396683304965683047207192982139922672025395099... is the root of the equation exp(1/r)/r + (1+exp(1/r))*LambertW(-exp(-1/r)/r) = 0, and c = 0.2796968489586733500739737080739303725411427162653658... . - _Vaclav Kotesovec_, May 07 2014
%p A122400 A122399 := proc(n) option remember ; add( combinat[stirling2](n,k)*k^n*k!,k=0..n) ; end: A122400 := proc(n) option remember ; add( combinat[stirling1](n,k)*A122399(k),k=0..n)/n! ; end: for n from 0 to 30 do printf("%d, ",A122400(n)) ; od ; # _R. J. Mathar_, May 18 2007
%t A122400 max = 17; CoefficientList[ Series[ 1 + Sum[ ((1 + x)^n - 1)^n, {n, 1, max}], {x, 0, max}], x] (* _Jean-François Alcover_, Mar 26 2013, after _Vladeta Jovovic_ *)
%Y A122400 Cf. A104602, A220353, A301581, A301582, A301583, A301584.
%K A122400 easy,nonn
%O A122400 0,3
%A A122400 _Vladeta Jovovic_, Aug 31 2006
%E A122400 More terms from _R. J. Mathar_, May 18 2007