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%I #23 Nov 13 2022 17:47:59
%S 1,0,1,1,3,7,20,59,184,593,1964,6642,22845,79667,281037,1001092,
%T 3595865,13009673,47366251,173415176,638044203,2357941142,8748646386,
%U 32576869203,121701491701,456012458965,1713339737086
%N Expansion of 1/(1-x^2*c(x)), c(x) the g.f. of A000108.
%C Diagonal sums of A106566.
%H G. C. Greubel, <a href="/A132364/b132364.txt">Table of n, a(n) for n = 0..1000</a>
%H Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL11/Barry/barry594.html">A note on Krawtchouk Polynomials and Riordan Arrays</a>, JIS 11 (2008) 08.2.2.
%H George Beck and Karl Dilcher, <a href="https://arxiv.org/abs/2106.10400">A Matrix Related to Stern Polynomials and the Prouhet-Thue-Morse Sequence</a>, arXiv:2106.10400 [math.CO], 2021.
%F a(0)=1, a(n) = Sum_{k=0..floor(n/2)} (k/(n-k))*C(2n-3k-1,n-2k)), n>0 .
%F G.f.: (2-x-x*sqrt(1-4*x))/(2-2*x+2*x^3). - _Philippe Deléham_, Feb 24 2013
%F 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
%F a(n) ~ 2^(2*n + 2) / (49 * sqrt(Pi) * n^(3/2)). - _Vaclav Kotesovec_, Nov 13 2022
%t 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 *)
%o (PARI) c(x) = (1 - sqrt(1 - 4*x)) / (2*x); \\ A000108
%o my(x='x+O('x^30)); Vec(1/(1-x^2*c(x))) \\ _Michel Marcus_, Nov 13 2022
%Y Cf. A000108, A030238, A106566
%K nonn
%O 0,5
%A _Philippe Deléham_, Nov 08 2007
%E Typo in a(n) term corrected _Johannes W. Meijer_, Sep 13 2010
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