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A174171 A generalized Chebyshev transform of the Motzkin numbers A001006. 1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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

LINKS

Table of n, a(n) for n=0..26.

P. 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.

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

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 *)

CROSSREFS

Cf. A001006.

Sequence in context: A328038 A107840 A046736 * A262042 A227644 A074188

Adjacent sequences:  A174168 A174169 A174170 * A174172 A174173 A174174

KEYWORD

easy,nonn

AUTHOR

Paul Barry, Mar 10 2010

STATUS

approved

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Last modified August 10 04:38 EDT 2020. Contains 336368 sequences. (Running on oeis4.)