A034171 Related to triple factorial numbers A007559(n+1).

1, 6, 42, 315, 2457, 19656, 160056, 1320462, 11003850, 92432340, 781473420, 6642524070, 56716936290, 486145168200, 4180848446520, 36059817851235, 311811366125385, 2702365173086670, 23467908082068450, 204170800313995515, 1779202688450532345, 15527587099204645920

Offset: 0

Author

Wolfdieter Lang

Keywords

• easy
• nonn

Comments

Working with an offset of 1, we conjecture a(p*n) = a(n) (mod p^2) for prime p = 1 (mod 3) and all positive integers n except those n of the form n = m*p + k for 0 <= m <= (p-1)/3 and 1 <= k <= (p-1)/3. Cf. A298799, A004981 and A004982. - Peter Bala, Dec 23 2019

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..1050
• Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), Article 00.2.4.
• Elżbieta Liszewska and Wojciech Młotkowski, Some relatives of the Catalan sequence, arXiv:1907.10725 [math.CO], 2019.

Crossrefs

Cf. A007559, A025748, A034164, A035529.

Cf. A298799, A004981, A004982, A004987, A073005.

Programs

Mathematica

CoefficientList[Series[(-1 + (1 - 9 x)^(-1/3))/(3 x), {x, 0, 19}], x] (* Michael De Vlieger, Oct 13 2019 *)

Formula

a(n) = 3^n*A007559(n+1)/(n+1)! where A007559(n+1)=(3*n+1)!!!.

G.f.: (-1+(1-9*x)^(-1/3))/(3*x).

a(n) = A035529(n+1, 1) (first column of triangle).

Convolution of A004987(n) with A025748(n+1), n >= 0.

From R. J. Mathar, Jan 28 2020: (Start)

D-finite with recurrence: (n+1)*a(n) + 3*(-3*n-1)*a(n-1) = 0.

G.f.: (1F0(1/3;;9*x)-1)/(3*x). (End)

Sum_{n>=0} 1/a(n) = 3/8 + 3*sqrt(3)*Pi/32 + 9*log(3)/32. - Amiram Eldar, Dec 22 2022

a(n) ~ 3^(2*n+1) * n^(-2/3) / Gamma(1/3). - Amiram Eldar, Aug 19 2025