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%I #16 Nov 02 2023 10:43:32
%S 1,3,12,57,294,1578,8658,48177,270774,1533450,8736432,50016090,
%T 287497380,1658174352,9591422286,55618701057,323225066790,
%U 1882009941570,10976834700792,64119701075886,375057555388884,2196539772794172,12878508015774468
%N Binomial transform of A126568, second binomial transform of A026641.
%C The Hankel transform of this sequence is 3^n (see A000244).
%H Isaac DeJager, Madeleine Naquin, Frank Seidl, <a href="https://www.valpo.edu/mathematics-statistics/files/2019/08/Drube2019.pdf">Colored Motzkin Paths of Higher Order</a>, VERUM 2019.
%F Conjecture: 2*n*a(n) + (-19*n+12)*a(n-1) + 6*(8*n-11)*a(n-2) + 36*(-n+2)*a(n-3) = 0. - _R. J. Mathar_, Jun 30 2013
%F a(n) ~ 2^(n + 1/2) * 3^(n - 1/2) / sqrt(Pi*n). - _Vaclav Kotesovec_, Nov 02 2023
%t CoefficientList[Series[(1 + 3*Sqrt[-1 + 2*x] / Sqrt[-1 + 6*x])/(4 - 6*x), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Nov 02 2023 *)
%Y Row sums of triangle in A124575.
%K nonn
%O 0,2
%A _Philippe Deléham_, Oct 08 2007
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