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%I #11 Aug 02 2021 15:50:19
%S 1,1,4,22,149,1169,10272,99012,1032346,11526094,136755650,1714031312,
%T 22584475206,311597054110,4486616619986,67227958200996,
%U 1045724188868353,16849477086762701,280694278424099214,4826423610068933738,85527389275821664161,1559842051063534891301
%N a(n) = Sum_{k=0..n} Stirling2(n,k) * binomial(3*k,k) / (2*k + 1).
%C Stirling transform of A001764.
%H Michael De Vlieger, <a href="/A346764/b346764.txt">Table of n, a(n) for n = 0..525</a>
%F G.f.: Sum_{k>=0} ( binomial(3*k,k) / (2*k + 1) ) * x^k / Product_{j=0..k} (1 - j*x).
%t Table[Sum[StirlingS2[n, k] Binomial[3 k, k]/(2 k + 1), {k, 0, n}], {n, 0, 21}]
%t nmax = 21; CoefficientList[Series[Sum[(Binomial[3 k, k]/(2 k + 1)) x^k/Product[1 - j x, {j, 0, k}], {k, 0, nmax}], {x, 0, nmax}], x]
%t nmax = 21; CoefficientList[Series[HypergeometricPFQ[{1/3, 2/3}, {1, 3/2}, 27 (Exp[x] - 1)/4], {x, 0, nmax}], x] Range[0, nmax]!
%o (PARI) a(n) = sum(k=0, n, stirling(n, k, 2)*binomial(3*k, k)/(2*k + 1)); \\ _Michel Marcus_, Aug 02 2021
%Y Cf. A001764, A064856, A346765, A346766, A346767, A346768, A346769.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Aug 02 2021
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