login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Like A001142, the 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

Table of n, a(n) for n=1..12.

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^(2k-n-1), {k, 1, n}]

a(n) = product[product[(1-1/k)^-k, {k, 2, j}], {j, 1, n}]

Also,

a(1) = 1; a(n) = a(n-1)product[(1-1/k)^-k, {k, 2, n}]

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

EXAMPLE

For n=3, row 3 of A003506 = {3, 6, 3} and a[3]=54.

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}]

Also,

a[1] = 1; a[n_] := a[n - 1] Product[(1 - 1/k)^-k, {k, 2, n}]

CROSSREFS

Cf. A003506, A001142, A007318. Begin with the second value of A001142:

a(n-1) = (n-1)!A001142, n>=2

Sequence in context: A003955 A182264 A094154 * A125531 A095209 A107101

Adjacent sequences:  A168507 A168508 A168509 * A168511 A168512 A168513

KEYWORD

easy,nonn

AUTHOR

Harlan J. Brothers, Nov 27 2009

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified February 25 08:12 EST 2018. Contains 299646 sequences. (Running on oeis4.)