OFFSET
1,2
COMMENTS
LINKS
A. Bogomolny, Cut The Knot: Leibniz and Pascal Triangles
H. J. Brothers, Pascal's prism, The Mathematical Gazette, 96 (July 2012), 213-220.
FORMULA
a(n) = n!*Product_{k=1..n} k^(2k-n-1).
a(n) = Product_{j=1..n} Product_{k=2..j} ((1-1/k)^-k).
a(1) = 1; a(n) = a(n-1)*Product_{k=2..n} ((1-1/k)^-k).
a(n) ~ A^2 * exp(n^2/2 - 1/12) * n^(n/2 + 1/6) / (2*Pi)^(n/2), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Oct 22 2017
a(n) = Product_{k=0..n-1} (n-k)^(n-2k). - Peter Munn, Mar 07 2018
EXAMPLE
For n=3, row 3 of A003506 = {3, 6, 3} and a(3)=54.
a(5) = 5^5 * 4^3 * 3^1 * 2^-1 * 1^-3 = 5^5 * 3 * 2^5 = 300000. - Peter Munn, Mar 07 2018
MATHEMATICA
Table[n! Product[k^(2 k - n - 1), {k, 1, n}], {n, 1, 12}]
Table[Product[Product[(1 - 1/k)^-k, {k, 2, j}], {j, 1, n}], {n, 1, 12}]
(* or *)
a[1] = 1; a[n_] := a[n - 1] Product[(1 - 1/k)^-k, {k, 2, n}]
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Harlan J. Brothers, Nov 27 2009
STATUS
approved