A055602 - OEIS

A055602

Number of n X n binary matrices with no 0 rows or columns and with n+1 1's.

0, 4, 45, 432, 4200, 43200, 476280, 5644800, 71850240, 979776000, 14270256000, 221298739200, 3642807168000, 63465795993600, 1167099373440000, 22596613079040000, 459548157100032000, 9795631769763840000, 218413777784057856000, 5084765197089177600000

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OFFSET

1,2

LINKS

Robert Israel, Table of n, a(n) for n = 1..445

FORMULA

Number of m X n binary matrices with no zero rows or columns and with k=0..m*n ones is Sum_{i=0..n} (-1)^i*binomial(n, i)*a(m, n-i, k) where a(m, n, k)=Sum_{i=0..m} (-1)^i*binomial(m, i)*binomial((m-i)*n, k).

a(n) = n*(n-1)*(n+2)*n!/4. - Vladeta Jovovic, Mar 25 2006

From Robert Israel, May 04 2021: (Start)

E.g.f.: x^2*(4-x)/(2*(1-x)^2).

D-finite with recurrence 4*(n-2)*a(n) - n*(4*n+3)*a(n-1) - (n-1)^2*a(n-2) = 0. (End)

From Amiram Eldar, Oct 23 2025: (Start)

Sum_{n>=2} 1/a(n) = 49/3 - 4*e/3 + 2*(gamma - Ei(1))/3, where e = A001113, Ei(1) = A091725, and gamma = A001620.

Sum_{n>=2} (-1)^n/a(n) = 10*(gamma - Ei(-1))/3 - 8/(3*e) - 13/9, where Ei(-1) = -A099285. (End)

MAPLE

f:= n -> n*(n-1)*(n+2)*n!/4:

map(f, [$1..30]); # Robert Israel, May 04 2021

MATHEMATICA

a[n_] := n*(n-1)*(n+2)*n!/4; Array[a, 20] (* Amiram Eldar, Oct 23 2025 *)