A154604 - OEIS

%I #22 May 31 2024 05:47:40

%S 1,1,3,54,9720,26244000,1488034800000,2362404048480000000,

%T 135019896025206528000000000,347259290825980971841536000000000000,

%U 49121618545275670528799969525760000000000000000

%N Hankel transform of reduced tangent numbers.

%C Hankel transform of A002105 (with interpolated zeros).

%C Hankel transform of A154603.

%H G. C. Greubel, <a href="/A154604/b154604.txt">Table of n, a(n) for n = 0..34</a>

%H Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Barry3/barry84r2.html">A Note on Three Families of Orthogonal Polynomials defined by Circular Functions, and Their Moment Sequences</a>, Journal of Integer Sequences, Vol. 15 (2012), #12.7.2. - _N. J. A. Sloane_, Dec 27 2012

%F a(n) = Product_{k=1..n} C(k+1,2)^(n-k+1).

%F 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

%t Table[Product[(k*(k+1)/2)^(n - k + 1), {k, 1, n}], {n, 0, 12}] (* _Vaclav Kotesovec_, Nov 13 2022 *)

%o (PARI) a(n) = prod(k=1, n, binomial(k+1,2)^(n-k+1)); \\ _Michel Marcus_, Nov 13 2022

%o (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

%o (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

%Y Cf. A002105, A154603.

%K easy,nonn

%O 0,3

%A _Paul Barry_, Jan 12 2009