OFFSET
0,2
COMMENTS
Also the number of n X n (0,1) matrices modulo rows permutation (by symmetry this is the same as the number of (0,1) matrices modulo columns permutation), i.e., the number of equivalence classes where two matrices A and B are equivalent if one of them is the result of permuting the rows of the other. The total number of (0,1) matrices is in sequence A002416.
Row sums of A220886. - Geoffrey Critzer, Nov 20 2014
LINKS
Harry J. Smith, Table of n, a(n) for n = 0..59
FORMULA
a(n) = [x^n] 1/(1-x)^(2^n).
a(n) = (1/n!)*Sum_{k=0..n} ( (-1)^(n-k)*Stirling1(n, k)*2^(k*n) ). - Vladeta Jovovic, May 28 2004
a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(2^n+n,k) - Vladeta Jovovic, Jan 21 2008
a(n) = Sum_{k=0..n} Stirling1(n,k)*(2^n+n-1)^k/n!. - Vladeta Jovovic, Jan 21 2008
G.f.: A(x) = Sum_{n>=0} [ -log(1 - 2^n*x)]^n / n!. More generally, Sum_{n>=0} [ -log(1 - q^n*x)]^n/n! = Sum_{n>=0} C(q^n+n-1,n)*x^n ; also Sum_{n>=0} log(1 + q^n*x)^n/n! = Sum_{n>=0} C(q^n,n)*x^n. - Paul D. Hanna, Dec 29 2007
a(n) ~ 2^(n^2) / n!. - Vaclav Kotesovec, Jul 02 2016
a(n) = A163767(2^n). - Alois P. Heinz, Jun 12 2024
MAPLE
with(combinat): for n from 0 to 20 do printf(`%d, `, binomial(2^n+n-1, n)) od:
MATHEMATICA
Table[Binomial[2^n+n-1, n], {n, 0, 20}] (* Harvey P. Dale, Apr 19 2012 *)
PROG
(PARI) a(n)=binomial(2^n+n-1, n)
(PARI) {a(n)=polcoeff(sum(k=0, n, (-log(1-2^k*x +x*O(x^n)))^k/k!), n)} \\ Paul D. Hanna, Dec 29 2007
(PARI) a(n) = sum(k=0, n, stirling(n, k, 1)*(2^n+n-1)^k/n!); \\ Paul D. Hanna, Nov 20 2014
(Sage) [binomial(2^n +n-1, n) for n in (0..20)] # G. C. Greubel, Mar 14 2021
(Magma) [Binomial(2^n +n-1, n): n in [0..20]]; // G. C. Greubel, Mar 14 2021
(Python)
from math import comb
def A060690(n): return comb((1<<n)+n-1, n) # Chai Wah Wu, Jul 05 2024
CROSSREFS
KEYWORD
nonn
AUTHOR
Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 19 2001
EXTENSIONS
More terms from James A. Sellers, Apr 20 2001
Edited by N. J. A. Sloane, Mar 17 2008
STATUS
approved