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

1, 1, 2, 4, 10, 27, 78, 234, 722, 2274, 7280, 23617, 77466, 256481, 856016, 2876940, 9728090, 33072228, 112974592, 387580856, 1334821448, 4613225722, 15994465796, 55615889745, 193904367362, 677709772035, 2374027931492, 8333765738127

Offset: 0

Author

Paul Barry, Feb 20 2009

Keywords

• easy
• nonn

Comments

Image of the Catalan numbers A000108 by the Riordan array (1, x*(1-x^2)). Hankel transform is A006720(n+2).

Partial sums of A157002.

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

Apparently: number of Dyck n-paths with no descent length equal to twice the preceding ascent length. - David Scambler, May 11 2012

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Jean-Luc Baril, Daniela Colmenares, José L. Ramírez, Emmanuel D. Silva, Lina M. Simbaqueba, and Diana A. Toquica, Consecutive pattern-avoidance in Catalan words according to the last symbol, Univ. Bourgogne (France 2023).
• Paul Barry, Integer sequences from elliptic curves, arXiv:2306.05025 [math.NT], 2023.
• P. Gawrychowski and P. K. Nicholson, Encodings of Range Maximum-Sum Segment Queries and Applications, arXiv:1410.2847 [cs.DS], 2014-2015.

Crossrefs

Cf. A000108.

Programs

Magma

m:=30; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!( (1-Sqrt(1-4*x*(1-x^2)))/(2*x*(1-x^2)) )); // G. C. Greubel, Feb 26 2019

Mathematica

CoefficientList[Series[(1-Sqrt[1-4*x*(1-x^2)])/(2*x*(1-x^2)),{x,0,20}],x] (* Vaclav Kotesovec, Jan 27 2015 *)

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

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

Formula

G.f.: c(x*(1-x^2)) where c(x) is the g.f. of A000108;

a(n) = Sum_{k=0..n} (-1)^((n-k)/2)*(1+(-1)^(n-k))*C(k,floor((n-k)/2))*A000108(k)/2.

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

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

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