A174169 A generalized Chebyshev transform of the Motzkin numbers A001006.

1, 1, -1, -2, 0, 0, -3, 1, 8, 1, 1, 26, 7, -51, -3, 0, -264, -186, 348, -120, -285, 2697, 2871, -2304, 3393, 8029, -25795, -36872, 16108, -60010, -159683, 213795, 413712, -181857, 833779, 2669534, -1272977, -4030235, 3611168, -9145271, -39467427

Offset

0,4

Comments

Hankel transform is the (1,3) Somos-4 sequence A174170.

Links

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Formula

G.f.: (1-x+3x^2-sqrt(1-2x+3x^2-6x^3+9x^4))/(2x^2)=(1/(1+3x))*M(x/(1+3x^2)), M(x) the g.f. of A010006;

a(n) = Sum_{k=0..floor(n/2)} (-3)^k*A001006(n-2k).

Conjecture: (n+2)*a(n) -(2*n+1)*a(n-1) +3*(n-1)*a(n-2) +3*(5-2*n)*a(n-3) +9*(n-4)*a(n-4)=0. - R. J. Mathar, Sep 30 2012

From Seiichi Manyama, Oct 17 2025: (Start)

G.f. A(x) satisfies A(x) = (1+x^2*A(x)^2) / (1-x+3*x^2).

a(0) = a(1) = 1; a(n) = a(n-1) - 3*a(n-2) + Sum_{k=0..n-2} a(k) * a(n-2-k). (End)

Crossrefs

Cf. A001006, A004148, A128720, A174171, A346503.

Sequence in context: A077888 A319668 A167634 * A248174 A125095 A113411

Adjacent sequences: A174166 A174167 A174168 * A174170 A174171 A174172

Keyword

easy,sign

Author

Paul Barry, Mar 10 2010

Status

approved