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a(n) = 2^(n*(n+1)/2)*A055209(n).
4

%I #20 Sep 08 2022 08:45:13

%S 1,2,32,9216,84934656,39137889484800,1298501242170900480000,

%T 4221953171600428240994304000000,

%U 1757091687362157576297527910177177600000000

%N a(n) = 2^(n*(n+1)/2)*A055209(n).

%C This sequence is the Hankel transform (see A001906 for definition) of A000629, A000670 (Fubini numbers) and A052841.

%H G. C. Greubel, <a href="/A091804/b091804.txt">Table of n, a(n) for n = 0..30</a>

%H P. Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Barry7/barry172.html">Eulerian Polynomials as Moments, via Exponential Riordan Arrays</a>, J. Int. Seq. 14 (2011) # 11.9.5.

%H P. 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=0..n} (2(k+1)*floor((2k+3)/2))^(n-k). - _Paul Barry_, Mar 30 2010

%t Table[Product[(2(k+1)Floor[(2k+3)/2])^(n-k),{k,0,n}],{n,0,10}] (* _Harvey P. Dale_, Aug 10 2011 *)

%o (Magma) [(&*[(2*(k+1)*Floor((2*k+3)/2))^(n-k): k in [0..n]]): n in [0..10]]; // _G. C. Greubel_, Oct 14 2018

%o (PARI) for(n=0,10, print1(prod(k=0,n, (2*(k+1)*floor((2*k+3)/2))^(n-k)), ", ")) \\ _G. C. Greubel_, Oct 14 2018

%Y Cf. A000629, A000670, A001906, A052841, A055209.

%K nonn

%O 0,2

%A _Philippe Deléham_, Mar 07 2004