A214589 -id:A214589 - cp's OEIS frontend

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

R. H. Hardin, Jul 22 2012

			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

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

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

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

/* 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

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

A386649 Product of first n central trinomial coefficients (A002426) for n > 0 with a(0) = 1.

Original entry on oeis.org

1, 1, 3, 21, 399, 20349, 2869209, 1127599137, 1248252244659, 3918263795984601, 35080215765450132753, 899912775031092255512709, 66403663756769266442027284401, 14140062564030204365431731967633341, 8713488333644640745496899895218790824407

Offset: 0

Paul D. Hanna, Aug 08 2025

nonn

Conjecture: a(n) = A214589(n) - 2 for n >= 1, where A214589(n) is the number of n X n X n triangular 0..2 arrays with every horizontal row having the same average value.

			The central trinomial coefficients A002426(n) = [x^n] (1 + x + x^2)^n for n >= 0 begin [1, 1, 3, 7, 19, 51, 141, 393, 1107, 3139, ...], where a(n) equals the product of the terms A002426(0) through A002426(n).

Paul D. Hanna, Table of n, a(n) for n = 0..70

Cf. A214589, A386650, A342177, A003046, A002426.

Table[Product[3^k * Hypergeometric2F1[1/2, -k, 1, 4/3], {k, 0, n}], {n, 0, 15}] (* Vaclav Kotesovec, Aug 09 2025 *)

{a(n) = prod(k=0,n, polcoef((1 + x + x^2)^k, k) )}
for(n=0,15,print1(a(n),", "))

a(n) = Product_{k=0..n} A002426(k) for n >= 0.

a(n) ~ c * 3^((n-1)*(n+3)/2) * exp(n/2) / (2^(n - 3/4) * Pi^(n/2 - 1/4) * n^(n/2 + 7/16)), where c = 1.123782729130753266489882099159237662230713685736... - Vaclav Kotesovec, Aug 09 2025

cp's OEIS Frontend

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

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