A132364 Expansion of 1/(1-x^2*c(x)), c(x) the g.f. of A000108.

1, 0, 1, 1, 3, 7, 20, 59, 184, 593, 1964, 6642, 22845, 79667, 281037, 1001092, 3595865, 13009673, 47366251, 173415176, 638044203, 2357941142, 8748646386, 32576869203, 121701491701, 456012458965, 1713339737086

Offset

0,5

Comments

Diagonal sums of A106566.

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Paul Barry, A note on Krawtchouk Polynomials and Riordan Arrays, JIS 11 (2008) 08.2.2.

George Beck and Karl Dilcher, A Matrix Related to Stern Polynomials and the Prouhet-Thue-Morse Sequence, arXiv:2106.10400 [math.CO], 2021.

Formula

a(0)=1, a(n) = Sum_{k=0..floor(n/2)} (k/(n-k))*C(2*n-3*k-1,n-2*k), n>0.

G.f.: (2-x-x*sqrt(1-4*x))/(2-2*x+2*x^3). - Philippe Deléham, Feb 24 2013

Conjecture: +(-n+1)*a(n) +(5*n-11)*a(n-1) +2*(-2*n+5)*a(n-2) +(-n+1)*a(n-3) +2*(2*n-5)*a(n-4)=0. - R. J. Mathar, Aug 28 2015

a(n) ~ 2^(2*n + 2) / (49 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Nov 13 2022

Mathematica

a[0] := 1; a[n_] := Sum[(k/(n - k))*Binomial[2*n - 3*k - 1, n - 2*k], {k, 0, Floor[n/2]}]; Table[a[n], {n, 0, 25}] (* G. C. Greubel, Oct 19 2016 *)

Prog

(PARI) c(x) = (1 - sqrt(1 - 4*x)) / (2*x); \\ A000108
my(x='x+O('x^30)); Vec(1/(1-x^2*c(x))) \\ Michel Marcus, Nov 13 2022

Crossrefs

Cf. A000108, A030238, A106566

Sequence in context: A129429 A084204 A030238 * A110490 A132868 A357792

Adjacent sequences: A132361 A132362 A132363 * A132365 A132366 A132367

Keyword

nonn

Author

Philippe Deléham, Nov 08 2007

Extensions

Typo in a(n) term corrected Johannes W. Meijer, Sep 13 2010

Status

approved