A054688 Number of nonnegative integer n X n matrices with sum of elements equal to n; polynomial symmetric functions of matrix of order n.

1, 1, 10, 165, 3876, 118755, 4496388, 202927725, 10639125640, 635627275767, 42634215112710, 3172596834321200, 259398433286078100, 23116565732981832150, 2230164446387219893320, 231574204669402103059965, 25751746463640423324267024, 3053419608195531383028424575

Offset: 0

Author

Vladeta Jovovic, Apr 19 2000

Keywords

• easy
• nonn

Comments

Conjecture: a(n) equals the sum of squares of the number of semistandard Young tableaux over all partitions of n. - Wouter Meeussen, Feb 02 2025

References

• E. R. Cavalcanti and M. A. Spohn, On the applicability of mobility metrics for user movement pattern recognition in MANETs, in Proceeding MobiWac '13 Proceedings of the 11th ACM international symposium on Mobility management and wireless access, Pages 123-130, ACM New York, NY, USA 2013, ISBN: 978-1-4503-2355-0 doi:10.1145/2508222.2508228

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..337

Crossrefs

Cf. A001700, A007716, A262030.

Programs

Maple

a:= n-> binomial(n*(n+1)-1, n):
seq(a(n), n=0..17);  # Alois P. Heinz, Oct 22 2021

Mathematica

Table[Binomial[n^2 + n - 1, n], {n, 0, 17}] (* Michael De Vlieger, Oct 05 2017 *)

PARI

a(n) = binomial(n^2+n-1, n); \\ Altug Alkan, Oct 03 2017

Formula

a(n) = C(n^2+n-1, n).

a(n) = [x^n] 1/(1 - x)^(n^2). - Ilya Gutkovskiy, Oct 03 2017

a(n) ~ exp(n + 1/2) * n^(n - 1/2) / sqrt(2*Pi). - Vaclav Kotesovec, Aug 06 2025

Extensions

a(15) corrected by Ilya Gutkovskiy, Oct 03 2017