A346767 a(n) = Sum_{k=0..n} Stirling2(n,k) * binomial(6*k,k) / (5*k + 1).

1, 1, 7, 70, 855, 11907, 182714, 3029040, 53565875, 1001599339, 19674910572, 404009742858, 8638256718929, 191702754433132, 4403979321915615, 104496256532120370, 2555972287817569101, 64340126437548435175, 1664318438781195696512, 44182488823505663971205

Offset: 0

Author

Ilya Gutkovskiy, Aug 02 2021

Keywords

• nonn

Comments

Stirling transform of A002295.

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..499

Crossrefs

Cf. A002295, A064856, A346764, A346765, A346766, A346768, A346769.

Programs

Mathematica

Table[Sum[StirlingS2[n, k] Binomial[6 k, k]/(5 k + 1), {k, 0, n}], {n, 0, 19}]
nmax = 19; CoefficientList[Series[Sum[(Binomial[6 k, k]/(5 k + 1)) x^k/Product[1 - j x, {j, 0, k}], {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 19; CoefficientList[Series[HypergeometricPFQ[{1/6, 1/3, 1/2, 2/3, 5/6}, {2/5, 3/5, 4/5, 1, 6/5}, 46656 (Exp[x] - 1)/3125], {x, 0, nmax}], x] Range[0, nmax]!

PARI

a(n) = sum(k=0, n, stirling(n, k, 2)*binomial(6*k, k)/(5*k + 1)); \\ Michel Marcus, Aug 02 2021

Formula

G.f.: Sum_{k>=0} ( binomial(6*k,k) / (5*k + 1) ) * x^k / Product_{j=0..k} (1 - j*x).