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%I #17 Apr 06 2021 09:56:46
%S 1,4,28,384,10496,573440,62652416,13690208256,5982889443328,
%T 5229277301702656,9141181343655264256,31958984107701798174720,
%U 223467104335874481157308416,3125102257923487167715657908224
%N Hankel transform of A123164.
%H G. C. Greubel, <a href="/A180966/b180966.txt">Table of n, a(n) for n = 0..80</a>
%F a(n) = 2^C(n,2)*Sum_{k=0..floor(n/2)} C(n-k,k)*(-2)^k*4^(n-2*k).
%F a(n) = 2^C(n,2)*[x^n] (1/(1 - 4*x + 2*x^2)).
%F a(n) = 2^(2*n + ((n-1)*n)/2)*Hyper2F1([(1-n)/2, -n/2], [-n], 1/2) for n > 0. - _Peter Luschny_, Aug 02 2014
%F a(n) ~ 2^(n^2/2 - 1) * (1 + sqrt(2))^(n+1). - _Vaclav Kotesovec_, Feb 14 2021
%F a(n) = 2^(n^2/2)*ChebyshevU(n, sqrt(2)) = 2^(n*(n-1)/2)*A007070(n). - _G. C. Greubel_, Apr 06 2021
%t a[n_] := 2^Binomial[n, 2] Sum[Binomial[n-k, k] (-2)^k 4^(n-2k), {k, 0, n/2} ]; Table[a[n], {n, 0, 13}] (* _Jean-François Alcover_, Jun 17 2019 *)
%t Table[2^(n^2/2)*ChebyshevU[n, Sqrt[2]], {n,0,20}] (* _G. C. Greubel_, Apr 06 2021 *)
%o (Magma) [ 2^Binomial(n,2)*(&+[ (-1)^k*Binomial(n-k,k)*2^(2*n-3*k): k in [0..Floor(n/2)]]): n in [0..20]]; // _G. C. Greubel_, Apr 06 2021
%o (Sage) [2^(n^2/2)*chebyshev_U(n, sqrt(2)) for n in (0..20)] # _G. C. Greubel_, Apr 06 2021
%Y Cf. A007070, A123164.
%K easy,nonn
%O 0,2
%A _Paul Barry_, Sep 29 2010
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