A174171 A generalized Chebyshev transform of the Motzkin numbers A001006.

1, 1, 4, 8, 25, 65, 197, 571, 1753, 5351, 16746, 52626, 167547, 536559, 1732272, 5622960, 18357211, 60205319, 198323708, 655787680, 2176141555, 7244106347, 24185285341, 80960692691, 271685400443, 913784117809, 3079889039230

Offset: 0

Author

Paul Barry, Mar 10 2010

Keywords

• easy
• nonn

Comments

Hankel transform is the (1,8) Somos-4 sequence A097495(n+2).

Links

• Paul Barry, Invariant number triangles, eigentriangles and Somos-4 sequences, arXiv preprint arXiv:1107.5490 [math.CO], 2011.
• Paul Barry, Generalized Catalan recurrences, Riordan arrays, elliptic curves, and orthogonal polynomials, arXiv:1910.00875 [math.CO], 2019.

Crossrefs

Cf. A001006.

Programs

Mathematica

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 *)

Formula

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.

a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k) * 2^k * A001006(n-2k).

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

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

a(n) ~ 17^(1/4) * (3 + sqrt(17))^(n+1) / (sqrt(Pi) * n^(3/2) * 2^(n+2)). - Vaclav Kotesovec, Nov 11 2021