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%I #39 Nov 11 2021 16:58:49
%S 1,1,4,8,25,65,197,571,1753,5351,16746,52626,167547,536559,1732272,
%T 5622960,18357211,60205319,198323708,655787680,2176141555,7244106347,
%U 24185285341,80960692691,271685400443,913784117809,3079889039230
%N A generalized Chebyshev transform of the Motzkin numbers A001006.
%C Hankel transform is the (1,8) Somos-4 sequence A097495(n+2).
%H Paul Barry, <a href="http://arxiv.org/abs/1107.5490">Invariant number triangles, eigentriangles and Somos-4 sequences</a>, arXiv preprint arXiv:1107.5490 [math.CO], 2011.
%H Paul Barry, <a href="https://arxiv.org/abs/1910.00875">Generalized Catalan recurrences, Riordan arrays, elliptic curves, and orthogonal polynomials</a>, arXiv:1910.00875 [math.CO], 2019.
%F G.f.: (1-x-2*x^2-sqrt(1-2*x-7*x^2+4*x^3+4*x^4))/(2*x^2) = (1/(1-2*x))*M(x/(1-2*x^2)), M(x) the g.f. of A010006.
%F a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k) * 2^k * A001006(n-2k).
%F Conjecture: (n+2)*a(n) -(2*n+1)*a(n-1) +7*(1-n)*a(n-2) +2*(2*n-5)*a(n-3) +4*(n-4)*a(n-4)=0. - _R. J. Mathar_, Sep 30 2012
%F a(0) = a(1) = 1; a(n) = a(n-1) + 2 * a(n-2) + Sum_{k=0..n-2} a(k) * a(n-k-2). - _Ilya Gutkovskiy_, Nov 09 2021
%F a(n) ~ 17^(1/4) * (3 + sqrt(17))^(n+1) / (sqrt(Pi) * n^(3/2) * 2^(n+2)). - _Vaclav Kotesovec_, Nov 11 2021
%t Table[Sum[Binomial[n - k, k] 2^k * Hypergeometric2F1[(1 - #)/2, -#/2, 2, 4] &[n - 2 k], {k, 0, Floor[n/2]}], {n, 0, 26}] (* _Michael De Vlieger_, Feb 02 2017, after _Peter Luschny_ at A001006 *)
%Y Cf. A001006.
%K easy,nonn
%O 0,3
%A _Paul Barry_, Mar 10 2010
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