A014070 a(n) = binomial(2^n, n).

1, 2, 6, 56, 1820, 201376, 74974368, 94525795200, 409663695276000, 6208116950265950720, 334265867498622145619456, 64832175068736596027448301568, 45811862025512780638750907861652480, 119028707533461499951701664512286557511680

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

a(n) is the number of n X n (0,1) matrices with distinct rows modulo rows permutations. - Yuval Dekel (dekelyuval(AT)hotmail.com), Nov 13 2003

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..60

Crossrefs

Sequences of the form binomial(2^n +p*n +q, n): A136556 (0,-1), this sequence (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. A088229, A136393, A136555.

Programs

Magma

[Binomial(2^n, n): n in [0..25]]; // Vincenzo Librandi, Sep 13 2016

Maple

A014070:= n-> binomial(2^n,n); seq(A014070(n), n=0..20); # G. C. Greubel, Mar 14 2021

Mathematica

Table[Binomial[2^n,n],{n,0,20}] (* Vladimir Joseph Stephan Orlovsky, Mar 03 2011 *)

PARI

a(n)=binomial(2^n,n)

PARI

/* G.f. A(x) as Sum of Series: */
a(n)=polcoeff(sum(k=0,n,log(1+2^k*x +x*O(x^n))^k/k!),n) \\ Paul D. Hanna, Dec 28 2007

PARI

{a(n) = (1/n!) * sum(k=0,n, stirling(n, k, 1) * 2^(n*k) )}
for(n=0,20,print1(a(n),", ")) \\ Paul D. Hanna, Feb 05 2023

Sage

[binomial(2^n, n) for n in (0..20)] # G. C. Greubel, Mar 14 2021

Formula

G.f.: A(x) = Sum_{n>=0} log(1 + 2^n*x)^n / n!. - Paul D. Hanna, Dec 28 2007

a(n) = (1/n!) * Sum_{k=0..n} Stirling1(n, k) * 2^(n*k). - Paul D. Hanna, Feb 05 2023

From Vaclav Kotesovec, Jul 02 2016: (Start)

a(n) ~ 2^(n^2) / n!.

a(n) ~ 2^(n^2 - 1/2) * exp(n) / (sqrt(Pi) * n^(n+1/2)).

(End)