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A289147 Number of (n+1) X (n+1) binary matrices M with at most one 1 in each of the first n rows and each of the first n columns and M[n+1,n+1] = 0. 5
1, 5, 34, 286, 2840, 32344, 414160, 5876336, 91356544, 1542401920, 28075364096, 547643910400, 11389266525184, 251428006132736, 5869482147358720, 144413021660821504, 3733822274973040640, 101181690628832198656, 2867011297057247002624, 84764595415605494743040 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Number of marriage patterns between a labeled set X of n women and a labeled set Y of n men (all heterosexual): some couples can be formed where one partner is from X and the other from Y, some members of X and Y marry external (unlabeled) partners, and some do not marry.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..437

Eric Weisstein's World of Mathematics, Laguerre Polynomial

Wikipedia, Laguerre polynomials

Index entries for sequences related to Laguerre polynomials

FORMULA

E.g.f.: exp(4*x/(1-x))/(1-x).

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

a(n) = Sum_{i=0..n} (n-i)! * 4^i * binomial(n,i)^2.

a(n) = n! * Sum_{i=0..n} 4^i/i! * binomial(n,i).

a(n) = (2*n+3)*a(n-1)-(n-1)^2*a(n-2) for n>=2, a(n) = 4*n+1 for n<2.

a(n) = n! * Laguerre(n,-4) = n! * A160611(n)/A160612(n).

a(n) ~ exp(-2 + 4*sqrt(n) - n) * n^(n + 1/4) / 2 * (1 + 163/(96*sqrt(n))). - Vaclav Kotesovec, Nov 13 2017

Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(x) * Sum_{n>=0} 4^n * x^n / (n!)^2. - Ilya Gutkovskiy, Jul 17 2020

EXAMPLE

a(1) = 5:

[0 0]  [1 0]  [0 1]  [0 0]  [0 1]

[0 0]  [0 0]  [0 0]  [1 0]  [1 0] .

.

a(2) = 34:

[0 0 0]  [0 0 0]  [0 0 0]  [0 0 0]  [0 0 0]  [0 0 0]  [0 0 0]

[0 0 0]  [0 0 0]  [0 0 0]  [0 0 0]  [0 0 1]  [0 0 1]  [0 0 1]

[0 0 0]  [0 1 0]  [1 0 0]  [1 1 0]  [0 0 0]  [0 1 0]  [1 0 0]

.

[0 0 0]  [0 0 0]  [0 0 0]  [0 0 0]  [0 0 0]  [0 0 1]  [0 0 1]

[0 0 1]  [0 1 0]  [0 1 0]  [1 0 0]  [1 0 0]  [0 0 0]  [0 0 0]

[1 1 0]  [0 0 0]  [1 0 0]  [0 0 0]  [0 1 0]  [0 0 0]  [0 1 0]

.

[0 0 1]  [0 0 1]  [0 0 1]  [0 0 1]  [0 0 1]  [0 0 1]  [0 0 1]

[0 0 0]  [0 0 0]  [0 0 1]  [0 0 1]  [0 0 1]  [0 0 1]  [0 1 0]

[1 0 0]  [1 1 0]  [0 0 0]  [0 1 0]  [1 0 0]  [1 1 0]  [0 0 0]

.

[0 0 1]  [0 0 1]  [0 0 1]  [0 1 0]  [0 1 0]  [0 1 0]  [0 1 0]

[0 1 0]  [1 0 0]  [1 0 0]  [0 0 0]  [0 0 0]  [0 0 1]  [0 0 1]

[1 0 0]  [0 0 0]  [0 1 0]  [0 0 0]  [1 0 0]  [0 0 0]  [1 0 0]

.

[0 1 0]  [1 0 0]  [1 0 0]  [1 0 0]  [1 0 0]  [1 0 0]

[1 0 0]  [0 0 0]  [0 0 0]  [0 0 1]  [0 0 1]  [0 1 0]

[0 0 0]  [0 0 0]  [0 1 0]  [0 0 0]  [0 1 0]  [0 0 0]  .

MAPLE

a:= proc(n) option remember; `if`(n<2, 4*n+1,

      (2*n+3)*a(n-1)-(n-1)^2*a(n-2))

    end:

seq(a(n), n=0..25);

# second Maple program:

a:= n-> n-> n! * add(binomial(n, i)*4^i/i!, i=0..n):

seq(a(n), n=0..25);

# third Maple program:

a:= n-> n!* simplify(LaguerreL(n, -4), 'LaguerreL'):

seq(a(n), n=0..25);

MATHEMATICA

Table[n! LaguerreL[n, -4], {n, 0, 30}] (* Indranil Ghosh, Jul 06 2017 *)

PROG

(Python)

from mpmath import *

mp.dps=150

l=chop(taylor(lambda x:exp(4*x/(1-x))/(1-x), 0, 31))

print [int(fac(i)*l[i]) for i in range(len(l))] # Indranil Ghosh, Jul 06 2017

# or #

from mpmath import *

mp.dps=100

def a(n): return int(fac(n)*laguerre(n, 0, -4))

print [a(n) for n in range(31)] # Indranil Ghosh, Jul 06 2017

CROSSREFS

Column k=4 of A289192.

Cf.: A000142, A000165, A000302, A002720, A025167, A084771, A087912, A102773, A160611, A160612, A277382.

Sequence in context: A189488 A111557 A211794 * A284864 A208677 A259906

Adjacent sequences:  A289144 A289145 A289146 * A289148 A289149 A289150

KEYWORD

nonn

AUTHOR

Alois P. Heinz, Jun 26 2017

STATUS

approved

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Last modified March 6 20:37 EST 2021. Contains 341850 sequences. (Running on oeis4.)