A136556 a(n) = binomial(2^n - 1, n).

1, 1, 3, 35, 1365, 169911, 67945521, 89356415775, 396861704798625, 6098989894499557055, 331001552386330913728641, 64483955378425999076128999167, 45677647585984911164223317311276545, 118839819203635450208125966070067352769535, 1144686912178270649701033287538093722740144666625

Offset

0,3

Comments

Number of n x n binary matrices without zero rows and with distinct rows up to permutation of rows, cf. A014070.

Row 0 of square array A136555.

Links

G. C. Greubel, Table of n, a(n) for n = 0..50

Formula

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

a(n) = (1/n!)*Sum_{k=0..n} Stirling1(n,k) * (2^n-1)^k.

G.f.: Sum_{n>=0} log(1 + 2^n*x)^n / (n! * (1 + 2^n*x)).

a(n) ~ 2^(n^2)/n!. - Vaclav Kotesovec, Jul 02 2016

Example

G.f.: A(x) = 1 + x + 3*x^2 + 35*x^3 + 1365*x^4 + 169911*x^5 +...
A(x) = 1/(1+x) + log(1+2*x)/(1+2*x) + log(1+4*x)^2/(2!*(1+4*x)) + log(1+8*x)^3/(3!*(1+8*x)) + log(1+16*x)^4/(4!*(1+16*x)) + log(1+32*x)^5/(5!*(1+32*x)) +...

Maple

A136556:= n-> binomial(2^n-1, n); seq(A136556(n), n=0..20); # G. C. Greubel, Mar 14 2021

Mathematica

f[n_] := Binomial[2^n - 1, n]; Array[f, 12] (* Robert G. Wilson v *)

Prog

(PARI) {a(n) = binomial(2^n-1, n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) /* As coefficient of x^n in the g.f.: */
{a(n) = polcoeff( sum(i=0, n, 1/(1 + 2^i*x +x*O(x^n)) * log(1 + 2^i*x +x*O(x^n))^i/i!), n)}
for(n=0, 20, print1(a(n), ", "))
(Sage) [binomial(2^n -1, n) for n in (0..20)] # G. C. Greubel, Mar 14 2021
(Magma) [Binomial(2^n -1, n): n in [0..20]]; // G. C. Greubel, Mar 14 2021

Crossrefs

Sequences of the form binomial(2^n +p*n +q, n): this sequence (0,-1), A014070 (0,0), A136505 (0,1), A136506 (0,2), A060690 (1,-1), A132683 (1,0), A132684 (1,1), A132685 (2,0), A132686 (2,1), A132687 (3,-1), A132688 (3,0), A132689 (3,1).

Cf. A066384, A136555, A136557.

Sequence in context: A279377 A215582 A136525 * A006098 A320845 A012499

Adjacent sequences: A136553 A136554 A136555 * A136557 A136558 A136559

Keyword

nonn

Author

Paul D. Hanna, Jan 07 2008; Paul Hanna and Vladeta Jovovic, Jan 15 2008

Extensions

Edited by N. J. A. Sloane, Jan 26 2008

Status

approved