A357407 a(n) = coefficient of x^n, n >= 0, in A(x) = exp( Sum_{n>=1} A183204(n)*x^n/n ), where A183204 equals the central terms of triangle A181544.

1, 4, 32, 360, 4964, 78064, 1344020, 24708928, 477282794, 9580852360, 198322047840, 4209371498256, 91221481924426, 2011834246746792, 45039165331725264, 1021419638492387856, 23426910170090512779, 542666070296546760492, 12681393784980089971368

Offset

0,2

Comments

A183204(n) = Sum_{k=floor(n/2)..n} C(n,k)^2 * C(2*k,n) * C(n+k,k).

A183204(n) equals the coefficient of x^n in (1-x)^(3*n+1) * Sum_{k>=0} binomial(n+k-1,k)^3 * x^k, which is the central term of row n of triangle A181544.

Links

Paul D. Hanna, Table of n, a(n) for n = 0..500

Formula

a(n) ~ c * 3^(3*n) / n^(5/2), where c = 0.289447274610263555814082139782101227837126089347468995035938970190651243... - Vaclav Kotesovec, Mar 14 2023

Example

G.f.: A(x) = 1 + 4*x + 32*x^2 + 360*x^3 + 4964*x^4 + 78064*x^5 + 1344020*x^6 + 24708928*x^7 + 477282794*x^8 + 9580852360*x^9 + 198322047840*x^10 + ...
where
log(A(x)) = 4*x + 48*x^2/2 + 760*x^3/3 + 13840*x^4/4 + 273504*x^5/5 + 5703096*x^6/6 + 123519792*x^7/7 + 2751843600*x^8/8 + 62659854400*x^9/9 + ... + A183204(n)*x^n/n + ...

Prog

(PARI) {A183204(n) = sum(k=n\2, n, binomial(n, k)^2 * binomial(2*k, n) * binomial(n+k, k) )}
{a(n) = polcoeff( exp( sum(m=1, n, A183204(m)*x^m/m ) + x*O(x^n) ), n)}
for(n=0, 20, print1(a(n), ", "))

Crossrefs

Cf. A183204, A181544.

Sequence in context: A177750 A304911 A224993 * A319220 A347021 A051489

Adjacent sequences: A357404 A357405 A357406 * A357408 A357409 A357410

Keyword

nonn

Author

Paul D. Hanna, Oct 19 2022

Status

approved