A055601 Number of n X n binary matrices with no zero rows.

1, 1, 9, 343, 50625, 28629151, 62523502209, 532875860165503, 17878103347812890625, 2375680873491867011912191, 1255325460068093790930770843649, 2644211984585174742731315532085090303, 22235498641774645581443610453175918212890625

Offset: 0

Author

Vladeta Jovovic, Jun 01 2000

Keywords

• easy
• nonn

Comments

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

Examples

			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! +...
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
		

Links

• Kenny Lau, Table of n, a(n) for n = 0..57

Crossrefs

Cf. A048291, A092477, A136516, A303560.

Programs

Maple

a:= n-> mul(Stirling2(n+1, 2), j=1..n): seq(a(n), n=0..10); # Zerinvary Lajos, Jan 01 2009

Mathematica

Join[{1},Table[(2^n-1)^n,{n,16}]] (* Vladimir Joseph Stephan Orlovsky, Feb 14 2011 *)

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

Python

a = lambda n:((1<Kenny Lau, Jul 05 2016

Python

N = 58
base = 0
a = []
for i in range(N):
    a += [base**i]
    base = (base<<1)|1 #base = base*2+1
print(a)
# Kenny Lau, Jul 05 2016

Formula

a(n) = A092477(n, n) for n>0.

a(n) = (2^n - 1 )^n. - Avi Peretz (njk(AT)netvision.net.il), Apr 21 2001

a(n) = Sum_{k=0..n} (-1)^k*C(n, k)*2^((n-k)*n).

E.g.f.: A(x) = Sum_{n>=0} 2^(n^2) * exp(-2^n*x) * x^n/n!. - Paul D. Hanna, Jan 02 2008

O.g.f.: Sum_{n>=0} 2^(n^2)*x^n/(1 + 2^n*x)^(n+1). - Paul D. Hanna, Jan 20 2010

Sum_{n>=1} 1/a(n) = A303560. - Amiram Eldar, Nov 18 2020