A224735 G.f.: exp( Sum_{n>=1} binomial(2*n,n)^3 * x^n/n ).

1, 8, 140, 3616, 116542, 4316080, 175593800, 7640774080, 349626142909, 16632958651688, 816163494236860, 41069537125459360, 2110206360805542510, 110346590629125981872, 5857345961837113457864, 314962180518584299711424, 17128125582951726423704502, 940726748732537798295599280

Offset

0,2

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..550

Formula

Logarithmic derivative yields A002897.

a(n) ~ c * 64^n / (Pi^(3/2) * n^(5/2)), where c = exp(HypergeometricPFQ[{1, 1, 3/2, 3/2, 3/2}, {2, 2, 2, 2}, 1]/8) = 1.1954231783227587621013437413385356072684907293727694463636... - Vaclav Kotesovec, Jul 16 2026

Example

G.f.: A(x) = 1 + 8*x + 140*x^2 + 3616*x^3 + 116542*x^4 + 4316080*x^5 +...
where
log(A(x)) = 2^3*x + 6^3*x^2/2 + 20^3*x^3/3 + 70^3*x^4/4 + 252^3*x^5/5 + 924^3*x^6/6 + 3432^3*x^7/7 + 12870^3*x^8/8 +...+ A000984(n)^3*x^n/n +...

Mathematica

CoefficientList[Series[Exp[8*x*HypergeometricPFQ[{1, 1, 3/2, 3/2, 3/2}, {2, 2, 2, 2}, 64*x]], {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 27 2025 *)

Prog

(PARI) {a(n)=polcoeff(exp(sum(k=1, n, binomial(2*k, k)^3*x^k/k)+x*O(x^n)), n)}
for(n=0, 20, print1(a(n), ", "))

Crossrefs

Cf. A224732, A224734, A224736, A002897, A000984.

Sequence in context: A212442 A185248 A228867 * A090931 A367199 A376094

Adjacent sequences: A224732 A224733 A224734 * A224736 A224737 A224738

Keyword

nonn,changed

Author

Paul D. Hanna, Apr 16 2013

Status

approved