A154604 Hankel transform of reduced tangent numbers.
1, 1, 3, 54, 9720, 26244000, 1488034800000, 2362404048480000000, 135019896025206528000000000, 347259290825980971841536000000000000, 49121618545275670528799969525760000000000000000
Offset: 0
Links
- G. C. Greubel, Table of n, a(n) for n = 0..34
- Paul Barry, A Note on Three Families of Orthogonal Polynomials defined by Circular Functions, and Their Moment Sequences, Journal of Integer Sequences, Vol. 15 (2012), #12.7.2. - _N. J. A. Sloane_, Dec 27 2012
Programs
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Magma
[n eq 0 select 1 else (&*[(Binomial(k+1,2))^(n-k+1): k in [1..n]]): n in [0..15]]; // G. C. Greubel, May 30 2024
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Mathematica
Table[Product[(k*(k+1)/2)^(n - k + 1), {k, 1, n}], {n, 0, 12}] (* Vaclav Kotesovec, Nov 13 2022 *) -
PARI
a(n) = prod(k=1, n, binomial(k+1,2)^(n-k+1)); \\ Michel Marcus, Nov 13 2022
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SageMath
[product((binomial(k+1,2))^(n-k+1) for k in range(1,n+1)) for n in range(16)] # G. C. Greubel, May 30 2024
Formula
a(n) = Product_{k=1..n} C(k+1,2)^(n-k+1).
a(n) ~ n^(n^2 + 3*n + 7/3) * Pi^(n + 3/2) / (A^2 * 2^((n^2 - n - 3)/2) * exp(3*n^2/2 + 3*n - 1/6)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Nov 13 2022
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