Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #9 Aug 03 2021 15:04:08
%S 1,1,6,51,531,6331,83532,1195452,18316582,297727712,5099398853,
%T 91554269703,1715910362408,33457504204403,676778172939139,
%U 14168046060375184,306327815585165519,6827996259530724139,156654003923243040925,3694188118839057258940,89428870506038692255920
%N a(n) = Sum_{k=0..n} Stirling2(n,k) * binomial(5*k,k) / (4*k + 1).
%C Stirling transform of A002294.
%F G.f.: Sum_{k>=0} ( binomial(5*k,k) / (4*k + 1) ) * x^k / Product_{j=0..k} (1 - j*x).
%t Table[Sum[StirlingS2[n, k] Binomial[5 k, k]/(4 k + 1), {k, 0, n}], {n, 0, 20}]
%t nmax = 20; CoefficientList[Series[Sum[(Binomial[5 k, k]/(4 k + 1)) x^k/Product[1 - j x, {j, 0, k}], {k, 0, nmax}], {x, 0, nmax}], x]
%t nmax = 20; CoefficientList[Series[HypergeometricPFQ[{1/5, 2/5, 3/5, 4/5}, {1/2, 3/4, 1, 5/4}, 3125 (Exp[x] - 1)/256], {x, 0, nmax}], x] Range[0, nmax]!
%o (PARI) a(n) = sum(k=0, n, stirling(n, k, 2)*binomial(5*k, k)/(4*k + 1)); \\ _Michel Marcus_, Aug 03 2021
%Y Cf. A002294, A064856, A346764, A346765, A346767, A346768, A346769.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Aug 02 2021