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A122400 Number of square (0,1)-matrices without zero rows and with exactly n entries equal to 1. 25

%I

%S 1,1,4,31,338,4769,82467,1687989,39905269,1069863695,32071995198,

%T 1062991989013,38596477083550,1523554760656205,64961391010251904,

%U 2975343608212835855,145687881987604377815,7594435556630244257213

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

%H G. C. Greubel, <a href="/A122400/b122400.txt">Table of n, a(n) for n = 0..375</a>

%F a(n) = (1/n!)* Sum_{k=0..n} Stirling1(n,k)*A122399(k).

%F G.f.: Sum_{n>=0} ((1+x)^n - 1)^n. - _Vladeta Jovovic_, Sep 03 2006

%F G.f.: Sum_{n>=0} (1+x)^(n^2) / (1 + (1+x)^n)^(n+1). - _Paul D. Hanna_, Mar 23 2018

%F 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 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 max = 17; CoefficientList[ Series[ 1 + Sum[ ((1 + x)^n - 1)^n, {n, 1, max}], {x, 0, max}], x] (* _Jean-Fran├žois Alcover_, Mar 26 2013, after _Vladeta Jovovic_ *)

%Y Cf. A104602, A220353, A301581, A301582, A301583, A301584.

%K easy,nonn

%O 0,3

%A _Vladeta Jovovic_, Aug 31 2006

%E More terms from _R. J. Mathar_, May 18 2007

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Last modified March 6 20:37 EST 2021. Contains 341850 sequences. (Running on oeis4.)