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

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

Offset: 0

Author

Paul D. Hanna, Aug 08 2025

Keywords

• nonn

Comments

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.

Examples

			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).
		

Links

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

Crossrefs

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

Programs

Mathematica

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

PARI

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

Formula

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