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A168510
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Products across consecutive rows of the denominators of the Leibniz harmonic triangle (A003506).
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1
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1, 4, 54, 2304, 300000, 116640000, 133413966000, 444110104166400, 4267295479315169280, 117595223746560000000000, 9245836018244425723200000000, 2065215715357207851951980544000000
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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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.
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LINKS
| A. Bogomolny, Cut The Knot: Leibniz and Pascal Triangles
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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}]
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EXAMPLE
| For n=3, row 3 of A003506 = {3, 6, 3} and a[3]=54.
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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}]
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CROSSREFS
| Cf. A003506, A001142, A007318. Begin with the second value of A001142:
a(n-1) = (n-1)!A001142, n>=2
Sequence in context: A201731 A003955 A094154 * A125531 A095209 A107101
Adjacent sequences: A168507 A168508 A168509 * A168511 A168512 A168513
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KEYWORD
| easy,nonn
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AUTHOR
| Harlan J. Brothers (harlan(AT)brotherstechnology.com), Nov 27 2009
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