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%I #10 Feb 15 2024 15:21:41
%S 1,1,53,5186,663444,98703235,16179000550,2837251240021,
%T 522937525075783,100134345595461824,19762585810520535829,
%U 3997199042964419204924,825055790810846248226675,173231819660726985218760834,36906136513918240767383588700,7962139696794640558535530147729
%N G.f.: exp(Sum_{k>=1} (4*k)!/(4!*k!^4) * x^k/k).
%F G.f. A(x) = G(x)^(1/24), where G(x) is the g.f. for A333042.
%F a(n) ~ c * 4^(4*n)/n^(5/2), where c = exp(HypergeometricPFQ[{1, 1, 5/4, 3/2, 7/4}, {2, 2, 2, 2}, 1] / 256) / (24*sqrt(2)*Pi^(3/2)) = 0.005320414767134132512371690902604699480645296829596277834542636529157577...
%t CoefficientList[Series[Exp[Sum[(4*k)!/(4!*k!^4)*x^k/k, {k, 1, 20}]], {x, 0, 20}], x]
%t CoefficientList[Series[Exp[x*HypergeometricPFQ[{1, 1, 5/4, 3/2, 7/4}, {2, 2, 2, 2}, 256*x]], {x, 0, 20}], x]
%Y Cf. A008977, A082368, A229452, A370295, A333042.
%K nonn
%O 0,3
%A _Vaclav Kotesovec_, Feb 14 2024