A214590 -id:A214590 - 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

A386650 Product of first n quadrinomial coefficients (A005725) for n > 0 with a(0) = 1.

Original entry on oeis.org

1, 1, 3, 30, 930, 93930, 31560480, 35600221440, 136099646565120, 1776236487321381120, 79580723341459838319360, 12296654209275691297430868480, 6578267322410960919238807125534720, 12223446894741861497849104893155093176320, 79112201841847644246811045518121813092796661760

Offset: 0

Paul D. Hanna, Aug 08 2025

nonn

Conjecture: 2*a(n) = A214590(n) - 2 for n >= 1, where A214590(n) is the number of nXnXn triangular 0..3 arrays with every horizontal row having the same average value.

			The quadrinomial coefficients A005725(n) = [x^n] (1 + x + x^2 + x^3)^n for n >= 0 begin [1, 1, 3, 10, 31, 101, 336, 1128, 3823, ...], where a(n) equals the product of the terms A005725(0) through A005725(n).

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

Cf. A214590, A386649, A342177, A003046, A005725.

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

{a(n) = prod(k=0, n, polcoef((1 + x + x^2 + x^3)^k, k) )}
for(n=0, 15, print1(a(n), ", "))

a(n) = Product_{k=0..n} A005725(k) for n >= 0.

a(n) ~ c * exp(n/2) * (11 + 217/(6371 + 624*sqrt(78))^(1/3) + (6371 + 624*sqrt(78))^(1/3))^(-1 + n/2 + n^2/2) * ((39 + (4563 - 78*sqrt(78))^(1/3) + (4563 + 78*sqrt(78))^(1/3))/13)^(n/2) / (2^(-11/4 + 2*n + n^2) * 3^((-3 + 2*n + n^2)/2) * Pi^(n/2 + 1/4) * n^((4290 - 1421*78^(2/3)/(804726 - 73709*sqrt(78))^(1/3) - (78*(804726 - 73709*sqrt(78)))^(1/3) + 4056*n)/8112)), where c = 0.77060824350557924602665408964165291884080801923663... - 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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