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Expansion of c(3*x^2)/(1-x*c(3*x^2)), c(x) the g.f. of A000108.
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%I #28 Aug 06 2024 04:48:12

%S 1,1,4,7,28,58,232,523,2092,4966,19864,48838,195352,492724,1970896,

%T 5068915,20275660,52955950,211823800,560198962,2240795848,5987822380,

%U 23951289520,64563867454,258255469816,701383563388,2805534253552

%N Expansion of c(3*x^2)/(1-x*c(3*x^2)), c(x) the g.f. of A000108.

%C Hankel transform is 3^C(n+1,2) = A047656(n+1).

%C Series reversion of x*(1+x)/(1+2*x+4*x^2).

%H G. C. Greubel, <a href="/A128386/b128386.txt">Table of n, a(n) for n = 0..1000</a>

%H 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 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 G.f.: (sqrt(1-12*x^2)+2*x-1)/(2*x*(1-4*x)).

%F 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 a(n) = Sum_{k=0..floor(n/2)} C(n,n-k)*(n-2*k+1)*3^k/(n-k+1);

%F a(n) = Sum_{k=0..floor(n/2)} A009766(n-k,k)*3^k.

%F a(n) = Sum_{k=0..n} 3^k*A120730(n,n-k). - _Philippe Deléham_, Mar 03 2007

%F 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 A120730[n_, k_]:= If[n>2*k, 0, Binomial[n,k]*(2*k-n+1)/(k+1)];

%t A126386[n_]:= Sum[3^k*A120730[n, n-k], {k,0,n}];

%t Table[A126386[n], {n,0,50}] (* _G. C. Greubel_, Nov 07 2022 *)

%o (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 (SageMath)

%o def A120730(n, k): return 0 if (n>2*k) else binomial(n, k)*(2*k-n+1)/(k+1)

%o def A126386(n): return sum(3^k*A120730(n,n-k) for k in range(n+1))

%o [A126386(n) for n in range(51)] # _G. C. Greubel_, Nov 07 2022

%Y Cf. A000108, A009766, A047656, A120730.

%K easy,nonn

%O 0,3

%A _Paul Barry_, Feb 28 2007