A090412 A Chebyshev transform of 2^n.

1, 2, 3, 4, 6, 10, 15, 20, 30, 52, 78, 96, 144, 282, 423, 420, 630, 1660, 2490, 1304, 1956, 11332, 16998, -3896, -5844, 95240, 142860, -157160, -235740, 983610, 1475415, -2634300, -3951450, 11751660, 17627490, -38381160, -57571740, 152461740, 228692610

Offset

0,2

Links

Table of n, a(n) for n=0..38.

Formula

G.f.: A(x) = c(-x^2)/(1-2*x*c(-x^2)), c(x) g.f. of Catalan numbers A000108.

a(n) = Sum_{k=0..n} (k+1)*binomial(n, n/2-k/2)*(-1)^(n/2 - k/2)*(1 + (-1)^(n+k))*2^k/(n+k+2)).

Let M be a tridiagonal matrix with 1's in the superdiagonal, [1,0,0,0,...] in the main diagonal, and [1,-1,-1,-1,...] in the subdiagonal; and V = vector [1,0,0,0,...]. The sequence is generated as a left column using iterates of M^n*V. - Gary W. Adamson, Jun 08 2011

D-finite with recurrence 2*(n+1)*a(n) -3*(n+1)*a(n-1) +8(n-2)*a(n-2) +12*(2-n)*a(n-3)=0. - R. J. Mathar, Nov 09 2012

The o.g.f. A(x) = (1/x) * Series reversion of x*(1 + 2*x)/((1 + x)*(1 + 3*x)). - Peter Bala, Nov 07 2022

Prog

(PARI) c(x) = (1 - sqrt(1 - 4*x)) / (2*x);
my(x='x+O('x^40)); Vec(c(-x^2)/(1-2*x*c(-x^2))) \\ Michel Marcus, Feb 06 2022

Crossrefs

Cf. A000108, A086990, A114710.

Sequence in context: A288807 A097699 A086990 * A073028 A147788 A104977

Adjacent sequences: A090409 A090410 A090411 * A090413 A090414 A090415

Keyword

easy,sign

Author

Paul Barry, Dec 05 2003

Status

approved