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%I #40 Apr 30 2021 12:39:47
%S 1,5,34,286,2840,32344,414160,5876336,91356544,1542401920,28075364096,
%T 547643910400,11389266525184,251428006132736,5869482147358720,
%U 144413021660821504,3733822274973040640,101181690628832198656,2867011297057247002624,84764595415605494743040
%N 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.
%C 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.
%H Alois P. Heinz, <a href="/A289147/b289147.txt">Table of n, a(n) for n = 0..437</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LaguerrePolynomial.html">Laguerre Polynomial</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Laguerre_polynomials">Laguerre polynomials</a>
%H <a href="/index/La#Laguerre">Index entries for sequences related to Laguerre polynomials</a>
%F E.g.f.: exp(4*x/(1-x))/(1-x).
%F a(n) = Sum_{i=0..n} i! * (2^(n-i)*binomial(n,i))^2.
%F a(n) = Sum_{i=0..n} (n-i)! * 4^i * binomial(n,i)^2.
%F a(n) = n! * Sum_{i=0..n} 4^i/i! * binomial(n,i).
%F 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.
%F a(n) = n! * Laguerre(n,-4) = n! * A160611(n)/A160612(n).
%F a(n) ~ exp(-2 + 4*sqrt(n) - n) * n^(n + 1/4) / 2 * (1 + 163/(96*sqrt(n))). - _Vaclav Kotesovec_, Nov 13 2017
%F 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
%e a(1) = 5:
%e [0 0] [1 0] [0 1] [0 0] [0 1]
%e [0 0] [0 0] [0 0] [1 0] [1 0] .
%e .
%e a(2) = 34:
%e [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0]
%e [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 1] [0 0 1] [0 0 1]
%e [0 0 0] [0 1 0] [1 0 0] [1 1 0] [0 0 0] [0 1 0] [1 0 0]
%e .
%e [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 1] [0 0 1]
%e [0 0 1] [0 1 0] [0 1 0] [1 0 0] [1 0 0] [0 0 0] [0 0 0]
%e [1 1 0] [0 0 0] [1 0 0] [0 0 0] [0 1 0] [0 0 0] [0 1 0]
%e .
%e [0 0 1] [0 0 1] [0 0 1] [0 0 1] [0 0 1] [0 0 1] [0 0 1]
%e [0 0 0] [0 0 0] [0 0 1] [0 0 1] [0 0 1] [0 0 1] [0 1 0]
%e [1 0 0] [1 1 0] [0 0 0] [0 1 0] [1 0 0] [1 1 0] [0 0 0]
%e .
%e [0 0 1] [0 0 1] [0 0 1] [0 1 0] [0 1 0] [0 1 0] [0 1 0]
%e [0 1 0] [1 0 0] [1 0 0] [0 0 0] [0 0 0] [0 0 1] [0 0 1]
%e [1 0 0] [0 0 0] [0 1 0] [0 0 0] [1 0 0] [0 0 0] [1 0 0]
%e .
%e [0 1 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0]
%e [1 0 0] [0 0 0] [0 0 0] [0 0 1] [0 0 1] [0 1 0]
%e [0 0 0] [0 0 0] [0 1 0] [0 0 0] [0 1 0] [0 0 0] .
%p a:= proc(n) option remember; `if`(n<2, 4*n+1,
%p (2*n+3)*a(n-1)-(n-1)^2*a(n-2))
%p end:
%p seq(a(n), n=0..25);
%p # second Maple program:
%p a:= n-> n-> n! * add(binomial(n, i)*4^i/i!, i=0..n):
%p seq(a(n), n=0..25);
%p # third Maple program:
%p a:= n-> n!* simplify(LaguerreL(n, -4), 'LaguerreL'):
%p seq(a(n), n=0..25);
%t Table[n! LaguerreL[n, -4], {n, 0, 30}] (* _Indranil Ghosh_, Jul 06 2017 *)
%o (Python)
%o from mpmath import *
%o mp.dps=150
%o l=chop(taylor(lambda x:exp(4*x/(1-x))/(1-x), 0, 31))
%o print([int(fac(i)*l[i]) for i in range(len(l))]) # _Indranil Ghosh_, Jul 06 2017
%o # or #
%o from mpmath import *
%o mp.dps=100
%o def a(n): return int(fac(n)*laguerre(n, 0, -4))
%o print([a(n) for n in range(31)]) # _Indranil Ghosh_, Jul 06 2017
%Y Column k=4 of A289192.
%Y Cf.: A000142, A000165, A000302, A002720, A025167, A084771, A087912, A102773, A160611, A160612, A277382.
%K nonn
%O 0,2
%A _Alois P. Heinz_, Jun 26 2017
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