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%I #34 Feb 03 2024 10:15:56
%S 1,3,7,17,42,106,272,708,1865,4963,13323,36037,98123,268737,739833,
%T 2046207,5682915,15842505,44315637,124348275,349911204,987212856,
%U 2791964574,7913642086,22477090679,63964370301,182353459733,520735012027,1489362193002,4266018891562,12236183875496,35142703099692,101055137177563
%N Pairwise sums of general ballot numbers (A002026).
%H G. C. Greubel, <a href="/A102071/b102071.txt">Table of n, a(n) for n = 1..1000</a>
%H Gennady Eremin, <a href="https://arxiv.org/abs/2004.09866">Naturalized bracket row and Motzkin triangle</a>, arXiv:2004.09866 [math.CO], 2020. See Table 2.
%F G.f.: (4*x*(1+x))/(1-x+sqrt(1-2*x-3*x^2))^2.
%F a(n) = (1/n) * Sum_{j=0..n} ((binomial(j,n-1-j)+4*binomial(j,n-2-j) + 3*binomial(j,n-3-j))*binomial(n,j)). - _Vladimir Kruchinin_, Mar 08 2016
%F a(n) ~ 4*3^(n+1/2)/(sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Mar 08 2016
%F a(n) = A001006(n+1) - A001006(n-1). - _Gennady Eremin_, Sep 23 2021
%F D-finite with recurrence (n+3)*a(n) + (-3*n-5)*a(n-1) + (-n+3)*a(n-2) + 3*(n-3)*a(n-3) = 0. - _R. J. Mathar_, Nov 01 2021
%F From _Peter Bala_, Feb 02 2024: (Start)
%F a(n) = Sum_{k = 0..n} (-1)^(n-k)*binomial(n,k)*A002057(k).
%F G.f.: x/(1 + x)*c(x/(1 + x))^4, where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108. (End)
%t CoefficientList[Series[(4x(1+x))/(1-x+Sqrt[1-2x-3x^2])^2,{x,0,40}],x] (* _Harvey P. Dale_, Feb 26 2013 *)
%o (Maxima)
%o a(n):=1/n*sum((binomial(j,n-1-j)+4*binomial(j,n-2-j)+3*binomial(j,n-3-j))*binomial(n,j),j,0,n); /* _Vladimir Kruchinin_, Mar 08 2016 */
%o (PARI) z='z+O('z^66); Vec(4*z*(1+z)/(1-z+sqrt(1-2*z-3*z^2))^2) \\ _Joerg Arndt_, Mar 08 2016
%Y First differences of A005554. Partial sums of A026269. 3rd column of A348840.
%Y Cf. A000108, A001006, A002057.
%K nonn,easy
%O 1,2
%A _Ralf Stephan_, Dec 30 2004
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