A110058 Number of nonnegative integer matrices of order n for which all row and column sums equal n.

1, 1, 3, 55, 10147, 22069251, 602351808741, 215717608046511873, 1046591482728407939338275, 70417932475495769964322670258947, 66880713903767740581650957184096513655153, 909176713758393122455793478657031533216492953328933, 178876969166665269546249744608783223036842010760723370462856181, 514016665650183402309555825250370336139392333285719205357202846243695510965

Offset: 0

Author

Brendan McKay, Sep 04 2005

Keywords

• nonn

Comments

Computed by a method that involves summing a multivariate generating function over roots of unity.

Examples

			a(2) = 3 due to the matrices [1,1 | 1,1], [0,2 | 2,0] and [2,0 | 0,2].
		

Links

• E. R. Canfield, B. D. McKay, Asymptotic enumeration of integer matrices with large equal row and column sums, Combinatorica 30 (6) (2010) 655-680

Crossrefs

Cf. A058407, A058410, A058391.

Main diagonal of A257493 and A333901.

Programs

Sage

from sage.combinat.integer_matrices import IntegerMatrices
[IntegerMatrices([n]*n, [n]*n).cardinality() for n in (0..6)] # Freddy Barrera, Dec 27 2018

Formula

log a(n) = 2(log 2)*n^2 - n*(log n) - n*(log 4*Pi) + (log n) + O(1). - Igor Pak, May 15 2019

Extensions

a(0)=1 prepended by Alois P. Heinz, Apr 26 2015