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%I #39 Apr 07 2021 20:00:24
%S 1,1,9,343,50625,28629151,62523502209,532875860165503,
%T 17878103347812890625,2375680873491867011912191,
%U 1255325460068093790930770843649,2644211984585174742731315532085090303,22235498641774645581443610453175918212890625
%N Number of n X n binary matrices with no zero rows.
%C 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_, Jan 02 2008
%H Kenny Lau, <a href="/A055601/b055601.txt">Table of n, a(n) for n = 0..57</a>
%F a(n) = A092477(n, n) for n>0.
%F a(n) = (2^n - 1 )^n. - Avi Peretz (njk(AT)netvision.net.il), Apr 21 2001
%F a(n) = Sum_{k=0..n} (-1)^k*C(n, k)*2^((n-k)*n).
%F E.g.f.: A(x) = Sum_{n>=0} 2^(n^2) * exp(-2^n*x) * x^n/n!. - _Paul D. Hanna_, Jan 02 2008
%F O.g.f.: Sum_{n>=0} 2^(n^2)*x^n/(1 + 2^n*x)^(n+1). - _Paul D. Hanna_, Jan 20 2010
%F Sum_{n>=1} 1/a(n) = A303560. - _Amiram Eldar_, Nov 18 2020
%e 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! +...
%e 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 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. - _Paul D. Hanna_, Jan 02 2008
%p a:= n-> mul(Stirling2(n+1, 2), j=1..n): seq(a(n), n=0..10); # _Zerinvary Lajos_, Jan 01 2009
%t Join[{1},Table[(2^n-1)^n,{n,16}]] (* _Vladimir Joseph Stephan Orlovsky_, Feb 14 2011 *)
%o (PARI) a(n)=n!*polcoeff(sum(k=0,n,2^(k^2)*exp(-2^k*x)*x^k/k!),n) \\ _Paul D. Hanna_, Jan 02 2008
%o (Python) a = lambda n:((1<<n)-1)**n # _Kenny Lau_, Jul 05 2016
%o (Python)
%o N = 58
%o base = 0
%o a = []
%o for i in range(N):
%o ....a += [base**i]
%o ....base = (base<<1)|1 #base = base*2+1
%o print(a)
%o # _Kenny Lau_, Jul 05 2016
%Y Cf. A048291, A092477, A136516, A303560.
%K easy,nonn
%O 0,3
%A _Vladeta Jovovic_, Jun 01 2000
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