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A048291 Number of {0,1} n X n matrices with no zero rows or columns. 25
1, 1, 7, 265, 41503, 24997921, 57366997447, 505874809287625, 17343602252913832063, 2334958727565749108488321, 1243237913592275536716800402887, 2630119877024657776969635243647463625 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Number of relations on n labeled points such that for every point x there exists y and z such that xRy and zRx.

Also the number of edge covers in the complete bipartite graph K_{n,n}. - Eric W. Weisstein, Apr 24 2017

REFERENCES

Brendan McKay, Posting to sci.math.research, Jun 14 1999.

LINKS

T. D. Noe, Table of n, a(n) for n=0..32

H. Cheballah, S. Giraudo, R. Maurice, Combinatorial Hopf algebra structure on packed square matrices, arXiv preprint arXiv:1306.6605 [math.CO], 2013.

R. J. Mathar, The number of nXm matrices with at most k 1's in each row or column, (2014).

R. Tauraso, Edge cover time for regular graphs, JIS 11 (2008) 08.4.4

Eric Weisstein's World of Mathematics, Complete Bipartite Graph

Eric Weisstein's World of Mathematics, Edge Cover

FORMULA

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

E.g.f.: Sum((2^n-1)^n*exp((1-2^n)*x)*x^n/n!,n=0..infinity). a(n) = Sum(Sum((-1)^(i+j)*binomial(n,i)*binomial(n,j)*2^(i*j),j = 0 .. n),i = 0 .. n). - Vladeta Jovovic, Feb 23 2008

a(n) ~ 2^(n^2). - Vaclav Kotesovec, Jul 02 2014

MAPLE

seq(sum((-1)^(n+k)*binomial(n, k)*(2^k-1)^n, k=1..n), n=1..15); # Robert FERREOL, Mar 10 2017

MATHEMATICA

Flatten[{1, Table[Sum[Binomial[n, k]*(-1)^k*(2^(n-k)-1)^n, {k, 0, n}], {n, 1, 15}]}] (* Vaclav Kotesovec, Jul 02 2014 *)

PROG

(PARI) a(n)=sum(k=0, n, binomial(n, k)*(-1)^k*(2^(n-k)-1)^n)

(Python)

import math

f = math.factorial

def A048291(n): return sum([(f(n)/f(s)/f(n - s))*(-1)**s*(2**(n - s) - 1)**n for s in range(0, n+1)]) # Indranil Ghosh, Mar 14 2017

CROSSREFS

Cf. A054976, A104602, A283624.

Cf. A055601, A055599, A104601, A086193, A086206.

Diagonal of A183109.

Sequence in context: A129423 A290880 A231486 * A015089 A179565 A069449

Adjacent sequences:  A048288 A048289 A048290 * A048292 A048293 A048294

KEYWORD

nonn,easy,nice

AUTHOR

Joe Keane (jgk(AT)jgk.org)

STATUS

approved

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Last modified January 22 23:00 EST 2019. Contains 319365 sequences. (Running on oeis4.)