This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A128386 #28 Aug 06 2024 04:48:12
%S A128386 1,1,4,7,28,58,232,523,2092,4966,19864,48838,195352,492724,1970896,
%T A128386 5068915,20275660,52955950,211823800,560198962,2240795848,5987822380,
%U A128386 23951289520,64563867454,258255469816,701383563388,2805534253552
%N A128386 Expansion of c(3*x^2)/(1-x*c(3*x^2)), c(x) the g.f. of A000108.
%C A128386 Hankel transform is 3^C(n+1,2) = A047656(n+1).
%C A128386 Series reversion of x*(1+x)/(1+2*x+4*x^2).
%H A128386 G. C. Greubel, <a href="/A128386/b128386.txt">Table of n, a(n) for n = 0..1000</a>
%H A128386 Alin Bostan, <a href="https://citeseerx.ist.psu.edu/pdf/749aef4c6f3668e652b5074e5268346ccecc88c9">Computer Algebra for Lattice Path Combinatorics</a>, Slides, Séminaire de Combinatoire Ph. Flajolet, March 28 2013.
%H A128386 Alin Bostan, Andrew Elvey Price, Anthony John Guttmann, and Jean-Marie Maillard, <a href="https://arxiv.org/abs/2001.00393">Stieltjes moment sequences for pattern-avoiding permutations</a>, arXiv:2001.00393 [math.CO], 2020.
%F A128386 G.f.: (sqrt(1-12*x^2)+2*x-1)/(2*x*(1-4*x)).
%F A128386 a(n) = (1/(n+1))*Sum_{k=0..n+1} Sum_{j=0..k} C(n,k)*C(k,j)*C(2*n-2*k+j, n-2*k+j)*(-1)^(n+j)*2^(2*k-j).
%F A128386 a(n) = Sum_{k=0..floor(n/2)} C(n,n-k)*(n-2*k+1)*3^k/(n-k+1);
%F A128386 a(n) = Sum_{k=0..floor(n/2)} A009766(n-k,k)*3^k.
%F A128386 a(n) = Sum_{k=0..n} 3^k*A120730(n,n-k). - _Philippe Deléham_, Mar 03 2007
%F A128386 D-finite with recurrence (n+1)*a(n) - 4*(n+1)*a(n-1) + 12*(2-n)*a(n-2) + 48*(n-2)*a(n-3) = 0. - _R. J. Mathar_, Nov 14 2011
%t A128386 A120730[n_, k_]:= If[n>2*k, 0, Binomial[n,k]*(2*k-n+1)/(k+1)];
%t A128386 A126386[n_]:= Sum[3^k*A120730[n, n-k], {k,0,n}];
%t A128386 Table[A126386[n], {n,0,50}] (* _G. C. Greubel_, Nov 07 2022 *)
%o A128386 (Magma) R<x>:=PowerSeriesRing(Rationals(), 40); Coefficients(R!( (Sqrt(1-12*x^2)+2*x-1)/(2*x*(1-4*x)) )); // _G. C. Greubel_, Nov 07 2022
%o A128386 (SageMath)
%o A128386 def A120730(n, k): return 0 if (n>2*k) else binomial(n, k)*(2*k-n+1)/(k+1)
%o A128386 def A126386(n): return sum(3^k*A120730(n,n-k) for k in range(n+1))
%o A128386 [A126386(n) for n in range(51)] # _G. C. Greubel_, Nov 07 2022
%Y A128386 Cf. A000108, A009766, A047656, A120730.
%K A128386 easy,nonn
%O A128386 0,3
%A A128386 _Paul Barry_, Feb 28 2007