A157002 -id:A157002 - cp's OEIS frontend

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

Paul Barry, Feb 20 2009

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

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.

Cf. A000108.

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

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

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

((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

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

A377264 Consider the recurrence d(k) = (d(k-3)d(k-2) + 1)/(d(k-5)d(k-4)d(k-3)^2d(k-2)^2d(k-1)), with d(0..4) = {1,1,1,2,1}. a(n) = numerator(d(2n+1)).

Original entry on oeis.org

1, 2, 3, 14, 69, 413, 7222, 90211, 2577626, 127385577, 5092018073, 655664812074, 78618294607139, 13276948495989478, 5995083279193033837, 1895278734817024984181, 1542333923096758721461086, 1867485777936169465836858947, 2020248742951852823878208914098, 7078136335206254534330825538868049

Offset: 0

Thomas Scheuerle, Oct 22 2024

nonn

Consider the sequence s(k) with ordinary generating function: 1/(1-d(0)*x/(1-d(1)*x/(1-d(2)*x/(...)))), the Hankel sequence transform of s(k) is A006720 starting with the third term.
This is a special case of a more general theorem: Consider the sequence h(k) with generating function 1/(1-c(0)*x/(1-c(1)*x/(1-c(2)*x/(...)))), if c(2*k+1) = (c(2*k-2)*c(2*k-1) + t)/(c(2*k-4)*c(2*k-3)*c(2*k-2)^2*c(2*k-1)^2*c(2*k)) for all k with c(< 0) = 1, then the Hankel sequence transform of h(k) satisfies a Somos-4 A(1, t) recurrence.
If we would change the start condition into d(0..4) = {1,1,-1,-2,(5/2)}, the expansion of the continued fraction generating function would give us A171416, its Hankel sequence transform is again A006720. There exist infinitely many sequences with the same Hankel sequence transform.

Cf. A006720, A028940.

The Hankel transform is directly related to A006720: A157002, A157003, A160702, A171416, A173992, A173993, A254314.

d(n) = if(n<5, [1,1,1,2,1][n+1], (d(n-3)*d(n-2)+1)/(d(n-5)*d(n-4)*d(n-3)^2*d(n-2)^2*d(n-1)))
a(n) = numerator(d(2*n+1))

a(n) = -numerator(ellmul(ellinit([0, 3, -1, 2, 0]), [-1,0], 2*n+1)[1])

a(n) = A006720(n+1)*A006720(n+3).
denominator(d(2*n+1)) = A006720(n+2)^2.
-a(n)/A006720(n+3)^2 are the x-coordinates of (2*n+1) times [-1,0] on the curve y^2 - y = x^3 + 3*x^2 + 2*x. "Times" means here the multiplication under the elliptic group law.

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

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A377264 Consider the recurrence d(k) = (d(k-3)d(k-2) + 1)/(d(k-5)d(k-4)d(k-3)^2d(k-2)^2d(k-1)), with d(0..4) = {1,1,1,2,1}. a(n) = numerator(d(2n+1)).

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PARI

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cp's OEIS Frontend

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

Views

Author

Keywords

Comments

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Crossrefs

Programs

Magma

Mathematica

PARI

Sage

Formula

A377264 Consider the recurrence d(k) = (d(k-3)*d(k-2) + 1)/(d(k-5)*d(k-4)*d(k-3)^2*d(k-2)^2*d(k-1)), with d(0..4) = {1,1,1,2,1}. a(n) = numerator(d(2*n+1)).

Views

Author

Keywords

Comments

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Programs

PARI

PARI

Formula

A377264 Consider the recurrence d(k) = (d(k-3)d(k-2) + 1)/(d(k-5)d(k-4)d(k-3)^2d(k-2)^2d(k-1)), with d(0..4) = {1,1,1,2,1}. a(n) = numerator(d(2n+1)).