A061718 a(n) = (n*(n+1)/2)^n.

1, 9, 216, 10000, 759375, 85766121, 13492928512, 2821109907456, 756680642578125, 253295162119140625, 103510234140112521216, 50714860157241037295616, 29345269354638035222576971

Offset: 1

Author

Jason Earls, Jun 20 2001

Keywords

• easy
• nonn

Comments

a(n) is the number of n X n matrices with nonnegative integer entries such that every row sum equals 2. - Sharon Sela (sharonsela(AT)hotmail.com), May 08 2002

Resultant of the polynomials P(n,x) and Q(n,x) where P(n,x)=sum(k=1,n,k*(-x)^k) and Q(n,x)=x^n-1. - Benoit Cloitre, Jan 26 2003

a(n) is also the number of positive-volume, axis-aligned, n-dimensional rectangular solids that have vertices in the set {0,1,...,n}^n. Proof: If (M_1,...,M_n) is the corner with the maximum coordinate values for such a solid, then there are (M_1)*...*(M_n) possibilities for the corner with the minimum coordinate values. The sum over all possibilities for M_1, ..., M_n can be factored into the product of n sums; each of the n sums simplifies to n(n+1)/2. - Lee A. Newberg, Aug 31 2009

Links

• Harry J. Smith, Table of n, a(n) for n = 1..100

Crossrefs

Cf. A066300.

Programs

Maple

a:=n->mul(sum(j, j=0..n),k=1..n): seq(a(n), n=1..13); # Zerinvary Lajos, Jun 02 2007
a:=n->mul(binomial(n+2,2), k=0..n): seq(a(n), n=0..12); # Zerinvary Lajos, Oct 02 2007

Mathematica

Table[((n(n+1))/2)^n,{n,20}] (* Harvey P. Dale, Dec 09 2022 *)
With[{nn=15},#[[1]]^#[[2]]&/@Thread[{Accumulate[Range[nn]],Range[nn]}]] (* Harvey P. Dale, Dec 25 2023 *)

PARI

{ for (n=1, 100, write("b061718.txt", n, " ", (n*(n + 1)/2)^n) ) } \\ Harry J. Smith, Jul 26 2009

Formula

a(n) = Sum_{i=1..n,j=1..n,k=1..n,...} (i*j*k*...). E.g., a(2) = 9 because 1*1 + 1*2 + 2*1 + 2*2 = 9. - Ben Paul Thurston, Aug 15 2006

a(n) = [x^n] 1/(1 - (n*(n + 1)/2)*x). - Ilya Gutkovskiy, Oct 10 2017

a(n) ~ 2^(-n)*exp(1-1/(2*n))*n^(2*n). - Stefano Spezia, Jun 13 2025