A346769 - OEIS

%I #10 Aug 02 2021 15:51:46

%S 1,1,9,117,1849,33099,648683,13652529,304828941,7160371928,

%T 175882500852,4497024667232,119255943612372,3270580645588057,

%U 92537409967439493,2695752129992788115,80716475549045336327,2480352681613911495046,78120174740199126232258

%N a(n) = Sum_{k=0..n} Stirling2(n,k) * binomial(8*k,k) / (7*k + 1).

%C Stirling transform of A007556.

%H Michael De Vlieger, <a href="/A346769/b346769.txt">Table of n, a(n) for n = 0..488</a>

%F G.f.: Sum_{k>=0} ( binomial(8*k,k) / (7*k + 1) ) * x^k / Product_{j=0..k} (1 - j*x).

%t Table[Sum[StirlingS2[n, k] Binomial[8 k, k]/(7 k + 1), {k, 0, n}], {n, 0, 18}]

%t nmax = 18; CoefficientList[Series[Sum[(Binomial[8 k, k]/(7 k + 1)) x^k/Product[1 - j x, {j, 0, k}], {k, 0, nmax}], {x, 0, nmax}], x]

%t nmax = 18; CoefficientList[Series[HypergeometricPFQ[{1/8, 1/4, 3/8, 1/2, 5/8, 3/4, 7/8}, {2/7, 3/7, 4/7, 5/7, 6/7, 1, 8/7}, 16777216 (Exp[x] - 1)/823543], {x, 0, nmax}], x] Range[0, nmax]!

%o (PARI) a(n) = sum(k=0, n, stirling(n, k, 2)*binomial(8*k, k)/(7*k + 1)); \\ _Michel Marcus_, Aug 02 2021

%Y Cf. A007556, A064856, A346764, A346765, A346766, A346767, A346768.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Aug 02 2021