Search: id:A055601
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%I A055601
%S A055601 1,1,9,343,50625,28629151,62523502209,532875860165503,
%T A055601 17878103347812890625,2375680873491867011912191,
%U A055601 1255325460068093790930770843649
%N A055601 Number of n X n binary matrices with no zero rows.
%C A055601 More generally, Sum_{n>=0} m^n * q^(n^2) * exp(b*q^n*x) * x^n / n! = 
               Sum_{n>=0} (m*q^n + b)^n * x^n / n! for all q, m, b. - Paul D. Hanna 
               (pauldhanna(AT)juno.com), Jan 02 2008
%F A055601 a(n) = (2^n - 1 )^n. - Avi Peretz (njk(AT)netvision.net.il), Apr 21 2001
%F A055601 a(n)=Sum_{k=0..n} (-1)^k*C(n, k)*2^((n-k)*n).
%F A055601 E.g.f.: A(x) = Sum_{n>=0} 2^(n^2) * exp(-2^n*x) * x^n/n!. - Paul D. Hanna 
               (pauldhanna(AT)juno.com), Jan 02 2008
%e A055601 A(x) = 1 + x + 3^2*x^2/2! + 7^3*x^3/3! + 15^4*x^4/4! +... + (2^n-1)^n*x^n/
               n! +... A(x) = exp(-x) + 2*exp(-2x) + 2^4*exp(-4x)*x^2/2! + 2^9*exp(-8x)*x^3/
               3! +...+ 2^(n^2)*exp(-2^n*x)*x^n/n! +...
%e A055601 This is a special case of the more general statement: Sum_{n>=0} m^n 
               * F(q^n*x)^b * log( F(q^n*x) )^n / n! = Sum_{n>=0} x^n * [y^n] F(y)^(m*q^n 
               + b) where F(x) = exp(x), q=2, m=1, b=-1.
%p A055601 restart:with (combinat):a:=n->mul(stirling2(n,2), j=2..n): seq(a(n), 
               n=1..10);# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jan 
               01 2009]
%o A055601 (PARI) a(n)=n!*polcoeff(sum(k=0,n,2^(k^2)*exp(-2^k*x)*x^k/k!),n) - Paul 
               D. Hanna (pauldhanna(AT)juno.com), Jan 02 2008
%Y A055601 Cf. A048291.
%Y A055601 a(n) = A092477(n, n) for n>0.
%Y A055601 Cf. A136516.
%Y A055601 Sequence in context: A098652 A110695 A157589 this_sequence A012812 A119756 
               A063068
%Y A055601 Adjacent sequences: A055598 A055599 A055600 this_sequence A055602 A055603 
               A055604
%K A055601 easy,nonn
%O A055601 0,3
%A A055601 Vladeta Jovovic (vladeta(AT)eunet.rs), Jun 01 2000

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