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A128814
a(0)=1, a(n)= Product_{k=1..n} k*(k+1)/2+1.
3
1, 2, 8, 56, 616, 9856, 216832, 6288128, 232660736, 10702393856, 599334055936, 40155381747712, 3172275158069248, 291849314542370816, 30936027341491306496, 3743259308320448086016, 512826525239901387784192, 78975284886944813718765568, 13583749000554507959627677696, 2594496059105911020288886439936
OFFSET
0,2
COMMENTS
Partial products of A000124.
LINKS
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.
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
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 *)
PROG
(PARI) a(n) = if (n, prod(k=1, n, k*(k+1)/2+1), 1); \\ Michel Marcus, Mar 18 2023
CROSSREFS
Cf. A000124.
Sequence in context: A363589 A243953 A005439 * A108208 A203199 A348875
KEYWORD
easy,nonn
AUTHOR
Miklos Kristof, Apr 10 2007
EXTENSIONS
More terms from Michel Marcus, Mar 18 2023
STATUS
approved