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A168510
Products across consecutive rows of the denominators of the Leibniz harmonic triangle (A003506).
2
1, 4, 54, 2304, 300000, 116640000, 133413966000, 444110104166400, 4267295479315169280, 117595223746560000000000, 9245836018244425723200000000, 2065215715357207851951980544000000
OFFSET
1,2
COMMENTS
As in A001142, lim_{n->inf} (a(n)a(n+2))/a(n+1)^2 = e, demonstrating an underlying relation between A003506 and Pascal's triangle A007318. Unlike A001142, in this case the function is asymptotic from above.
LINKS
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
Cf. A003506, A001142, A007318. For n >= 1, a(n) = n!*A001142(n).
Sequence in context: A357511 A374893 A094154 * A324235 A125531 A095209
KEYWORD
easy,nonn
AUTHOR
Harlan J. Brothers, Nov 27 2009
STATUS
approved