A180966 Hankel transform of A123164.
1, 4, 28, 384, 10496, 573440, 62652416, 13690208256, 5982889443328, 5229277301702656, 9141181343655264256, 31958984107701798174720, 223467104335874481157308416, 3125102257923487167715657908224
Offset: 0
Links
- G. C. Greubel, Table of n, a(n) for n = 0..80
Programs
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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
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Mathematica
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 *) Table[2^(n^2/2)*ChebyshevU[n, Sqrt[2]], {n,0,20}] (* G. C. Greubel, Apr 06 2021 *) -
Sage
[2^(n^2/2)*chebyshev_U(n, sqrt(2)) for n in (0..20)] # G. C. Greubel, Apr 06 2021
Formula
a(n) = 2^C(n,2)*Sum_{k=0..floor(n/2)} C(n-k,k)*(-2)^k*4^(n-2*k).
a(n) = 2^C(n,2)*[x^n] (1/(1 - 4*x + 2*x^2)).
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
a(n) ~ 2^(n^2/2 - 1) * (1 + sqrt(2))^(n+1). - Vaclav Kotesovec, Feb 14 2021
a(n) = 2^(n^2/2)*ChebyshevU(n, sqrt(2)) = 2^(n*(n-1)/2)*A007070(n). - G. C. Greubel, Apr 06 2021