A342578 a(n) = n! * [x^n] (Sum_{j>=0} n^(j*(j+1)/2) * x^j/j!)^(1/n) for n > 0, a(0) = 1.

1, 1, 3, 199, 249337, 6062674201, 3653786369479951, 65709007885111803731947, 40564683796482484146182142025377, 969773549559254966290998252899999751714721, 999999990999996719397362087568018696141879478712251051, 49037072510879011742983689973641327840345400616866967292640434759551

Offset: 0

Author

Alois P. Heinz, Mar 15 2021

Keywords

• nonn

Comments

All terms are odd.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..35
• Richard Stanley, Proof of the general conjecture, MathOverflow, March 2021.

Crossrefs

Cf. A000217, A023813, A178315, A178319.

Main diagonal of A346061.

Programs

Maple

a:= n-> `if`(n>0, coeff(series(add(n^binomial(j+1, 2)*
         x^j/j!, j=0..n)^(1/n), x, n+1), x, n)*n!, 1):
seq(a(n), n=0..12);

Formula

a(n) == 1 (mod n*(n-1)) for n >= 2 (see "general conjecture" in A178319 and link to proof by Richard Stanley above).

a(n) ~ n^((n^2 + n - 2)/2). - Vaclav Kotesovec, Jul 15 2021