A346766 - 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