A214595 T(n,k) = number of n X n X n triangular 0..k arrays with every horizontal row having the same average value.

2, 3, 2, 4, 5, 2, 5, 8, 23, 2, 6, 13, 62, 401, 2, 7, 18, 157, 1862, 20351, 2, 8, 25, 312, 10177, 187862, 2869211, 2, 9, 32, 601, 33352, 3330677, 63120962, 1127599139, 2, 10, 41, 986, 103651, 20608352, 5495329427, 71200442882, 1248252244661, 2, 11, 50, 1619

Offset: 1

Author

R. H. Hardin, Jul 22 2012

Keywords

• nonn
• tabl
• changed

Examples

			Table starts
.2.....3......4.......5........6.........7.........8..........9.........10
.2.....5......8......13.......18........25........32.........41.........50
.2....23.....62.....157......312.......601.......986.......1619.......2426
.2...401...1862...10177....33352....103651....250042.....589763....1199614
.2.20351.187862.3330677.20608352.121537201.493575042.1877543213.5767190924
Some solutions for n = k = 4:
.....2........1........2........2........2........2........2........2
....3.1......0.2......2.2......3.1......2.2......1.3......4.0......4.0
...3.2.1....0.3.0....3.2.1....2.4.0....0.2.4....3.0.3....1.2.3....4.0.2
..2.2.3.1..2.1.0.1..1.2.4.1..4.2.2.0..1.4.3.0..4.0.2.2..3.2.3.0..4.0.4.0
		

Links

• R. H. Hardin, Table of n, a(n) for n = 1..1475

Crossrefs

Row 2 is A000982(n+1). Other rows: A214596, A214597, A214598.

Columns: A214589, A214590, A214591, A214592, A214593, A214594.

Programs

PARI

/* helper function  mult() gives multiplicity of a composition */
mult(p, L=1, m=(#p)!)={for(k=2,#p, p[k]!=p[k-1] && m\=(-L+L=k)!); m\(#p-L+1)!}
A214595(n, k)={sum(a=1,k, prod(L=2,n, my(c=0); forpart(p=L*a, c+=mult(p), [0,k], L); c))+1} \\ M. F. Hasler, Aug 21 2025

Formula

Empirical for row n:

n=1: a(k)=2*a(k-1)-a(k-2)

n=2: a(k)=2*a(k-1)-2*a(k-3)+a(k-4)

n=3: (order 12 antisymmetric)

n=4: (order 32 symmetric)

n=5: (order 84 symmetric)

T(n, k) = Sum_{s=0..k} Product_{L=2..n} NC(s*L, L, k), where NC(s, n, k) is the number of compositions of sum s with n parts between 0 and k. - M. F. Hasler, Aug 21 2025