A128814 a(0)=1, a(n)= Product_{k=1..n} k*(k+1)/2+1.

1, 2, 8, 56, 616, 9856, 216832, 6288128, 232660736, 10702393856, 599334055936, 40155381747712, 3172275158069248, 291849314542370816, 30936027341491306496, 3743259308320448086016, 512826525239901387784192, 78975284886944813718765568, 13583749000554507959627677696, 2594496059105911020288886439936

Offset: 0

Author

Miklos Kristof, Apr 10 2007

Keywords

• easy
• nonn

Comments

Partial products of A000124.

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..269
• Spencer J. Franks, Pamela E. Harris, Kimberly Harry, Jan Kretschmann, and Megan Vance, Counting Parking Sequences and Parking Assortments Through Permutations, arXiv:2301.10830 [math.CO], 2023.

Crossrefs

Cf. A000124.

Programs

Maple

a[0]:=1:for n from 1 to 20 do a[n]:=product(k*(k+1)/2+1,k=1..n) od: seq(a[n],n=0..20);

Mathematica

FoldList[Times,Accumulate[Range[0,20]]+1] (* Harvey P. Dale, Apr 21 2024 *)

PARI

a(n) = if (n, prod(k=1, n, k*(k+1)/2+1), 1); \\ Michel Marcus, Mar 18 2023

Formula

a(n) grows roughly like n*(n!)^2/2^n. [Corrected by Vaclav Kotesovec, Mar 18 2023]

G.f.: G(0)/(2*x^2) - 1/x^2 - 1/x, where G(k)= 1 + 1/(1 - x*(k^2-k+2)/(x*(k^2-k+2) + 2/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jul 18 2013

a(n) = -2^(-n-1)*Gamma(n+3/2+sqrt(-7)/2)*Gamma(n+3/2-sqrt(-7)/2)*sin((3+sqrt(-7))*Pi/2)/Pi. - Robert Israel, May 19 2014

a(n) ~ cosh(sqrt(7)*Pi/2) * n^(2*(n+1)) / (2^n * exp(2*n)). - Vaclav Kotesovec, Mar 18 2023

Extensions

More terms from Michel Marcus, Mar 18 2023