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A155051 Expansion of c(x^2)*(1+x)/(1-x), c(x) the g.f. of A000108.

%I A155051 #9 Sep 30 2017 23:52:37
%S A155051 1,2,3,4,6,8,13,18,32,46,88,130,262,394,823,1252,2682,4112,8974,13836,
%T A155051 30632,47428,106214,165000,373012,581024,1323924,2066824,4741264,
%U A155051 7415704,17110549,26805394,62163064,97520734,227165524
%N A155051 Expansion of c(x^2)*(1+x)/(1-x), c(x) the g.f. of A000108.
%C A155051 Row sums of A155050.
%C A155051 Conjecture: A000975(n) = A264784(a(n-1)) for n > 0. - _Reinhard Zumkeller_, Dec 04 2015
%H A155051 G. C. Greubel, <a href="/A155051/b155051.txt">Table of n, a(n) for n = 0..1000</a>
%F A155051 a(n) = 2*Sum_{k=0..n,} ( C(k/2)*(1+(-1)^k)/2 ) - C(n/2)*(1+(-1)^n)/2, C(n) = A000108;
%F A155051 a(n) = (C(n/2) + 2*Sum_{k=0..(n/2-1), C(k)})*(1+(-1)^n)/2 + Sum_{k=0..n/2, C(k)}*(1-(-1)^n), C(n) = A000108.
%F A155051 Conjecture: (n+2)*a(n) -2*a(n-1) +(-5*n+4)*a(n-2) +8*a(n-3) +4*(n-3)*a(n-4)=0. - _R. J. Mathar_, Feb 05 2015
%F A155051 Conjecture: -(n+2)*(n-3)*a(n) +(n^2-n-10)*a(n-1) +4*(n^2-4*n+5)*a(n-2) -4*(n-2)^2*a(n-3)=0. - _R. J. Mathar_, Feb 05 2015
%t A155051 A155051[n_] := 2*Sum[CatalanNumber[k/2]*(1 + (-1)^k)/2, {k, 0, n}] -
%t A155051 CatalanNumber[n/2]*(1 + (-1)^n)/2; Table[A155051[n], {n, 0, 50}] (* _G. C. Greubel_, Sep 30 2017 *)
%Y A155051 Cf. A000108, A155050, A000975, A264784.
%K A155051 easy,nonn
%O A155051 0,2
%A A155051 _Paul Barry_, Jan 19 2009