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

Author

Philippe Deléham, Nov 08 2007

Keywords

• nonn

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.

Crossrefs

Cf. A000108, A030238, A106566

Programs

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 *)

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

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

Extensions

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