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A157003 Transform of Catalan numbers whose Hankel transform gives the Somos-4 sequence.

%I A157003 #35 Jun 14 2023 17:18:55
%S A157003 1,1,2,4,10,27,78,234,722,2274,7280,23617,77466,256481,856016,2876940,
%T A157003 9728090,33072228,112974592,387580856,1334821448,4613225722,
%U A157003 15994465796,55615889745,193904367362,677709772035,2374027931492,8333765738127
%N A157003 Transform of Catalan numbers whose Hankel transform gives the Somos-4 sequence.
%C A157003 Image of the Catalan numbers A000108 by the Riordan array (1, x*(1-x^2)). Hankel transform is A006720(n+2).
%C A157003 Partial sums of A157002.
%C A157003 Empirical: number of Dyck n-paths that avoid any one of {UDUDD, UUDDD, UUDUD, UUUDD}. e.g. of the 5 Dyck 3-paths UUDUDD contains UDUDD so a(3)=4. Also, number of Dyck n-paths that avoid DUD that ends at height of form 3*k+1, or that avoid UDU that ends at height of form 3*k-1. e.g. of the 5 Dyck 3-paths UUDUDD contains DUD ending at height 1 so a(3)=4. - _David Scambler_, Mar 24 2011
%C A157003 Apparently: number of Dyck n-paths with no descent length equal to twice the preceding ascent length. - _David Scambler_, May 11 2012
%H A157003 G. C. Greubel, <a href="/A157003/b157003.txt">Table of n, a(n) for n = 0..1000</a>
%H A157003 Jean-Luc Baril, Daniela Colmenares, José L. Ramírez, Emmanuel D. Silva, Lina M. Simbaqueba, and Diana A. Toquica, <a href="http://jl.baril.u-bourgogne.fr/bacorasisito.pdf">Consecutive pattern-avoidance in Catalan words according to the last symbol</a>, Univ. Bourgogne (France 2023).
%H A157003 Paul Barry, <a href="https://arxiv.org/abs/2306.05025">Integer sequences from elliptic curves</a>, arXiv:2306.05025 [math.NT], 2023.
%H A157003 P. Gawrychowski and P. K. Nicholson, <a href="http://arxiv.org/abs/1410.2847">Encodings of Range Maximum-Sum Segment Queries and Applications</a>, arXiv:1410.2847 [cs.DS], 2014-2015.
%F A157003 G.f.: c(x*(1-x^2)) where c(x) is the g.f. of A000108;
%F A157003 a(n) = Sum_{k=0..n} (-1)^((n-k)/2)*(1+(-1)^(n-k))*C(k,floor((n-k)/2))*A000108(k)/2.
%F A157003 Conjecture: (n+1)*a(n) +(n+2)*a(n-1) +(-21*n+29)*a(n-2) +(3*n-16)*a(n-3) +40*(n-3)*a(n-4) +2*(-2*n+7)*a(n-5) +10*(-2*n+9)*a(n-6)=0. - _R. J. Mathar_, Nov 19 2014
%F A157003 Recurrence: (n+1)*a(n) = 2*(2*n-1)*a(n-1) + (n+1)*a(n-2) - 8*(n-2)*a(n-3) + 2*(2*n-7)*a(n-5). - _Vaclav Kotesovec_, Feb 01 2015
%F A157003 a(n) ~ sqrt(3-8*r) / (sqrt(Pi) * n^(3/2) * r^n), where r = 2*sin(arccos(-3^(3/2)/8)/3 - Pi/6)/sqrt(3). - _Vaclav Kotesovec_, Jun 05 2022
%t A157003 CoefficientList[Series[(1-Sqrt[1-4*x*(1-x^2)])/(2*x*(1-x^2)),{x,0,20}],x] (* _Vaclav Kotesovec_, Jan 27 2015 *)
%o A157003 (PARI) my(x='x+O('x^30)); Vec((1-sqrt(1-4*x*(1-x^2)))/(2*x*(1-x^2))) \\ _G. C. Greubel_, Feb 26 2019
%o A157003 (Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!( (1-Sqrt(1-4*x*(1-x^2)))/(2*x*(1-x^2)) )); // _G. C. Greubel_, Feb 26 2019
%o A157003 (Sage) ((1-sqrt(1-4*x*(1-x^2)))/(2*x*(1-x^2))).series(x, 30).coefficients(x, sparse=False) # _G. C. Greubel_, Feb 26 2019
%Y A157003 Cf. A000108.
%K A157003 easy,nonn
%O A157003 0,3
%A A157003 _Paul Barry_, Feb 20 2009