A064641 Unidirectional 'Delannoy' variation of the Boustrophedon transform applied to all 1's sequence: construct an array in which the first element of each row is 1 and subsequent entries are given by T(n,k) = T(n,k-1) + T(n-1,k-1) + T(n-1,k) + T(n-2,k-1). The last number in row n gives a(n).

1, 2, 7, 29, 133, 650, 3319, 17498, 94525, 520508, 2910895, 16487795, 94393105, 545337200, 3175320607, 18615098837, 109783526821, 650884962908, 3877184797783, 23193307022861, 139271612505361, 839192166392276

Offset

0,2

Comments

Also the number of paths from (0,0) to (n,n) not rising above y=x, using steps (1,0), (0,1), (1,1) and (2,1). For example, the 7 paths to (2,2) are dd, den, end, enen, Dn, eenn and edn, where e=(1,0), n=(0,1), d=(1,1) and D=(2,1). - Brian Drake, Aug 01 2007

For another interpretation as the number of walks of a certain type, see A223092 and the link below. - N. J. A. Sloane, Mar 29 2013

Hankel transform is 3^C(n+1,2). - Paul Barry, Jan 26 2009

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Paul Barry, On a transformation of Riordan moment sequences, arXiv:1802.03443 [math.CO], 2018.

Paul Barry, Generalized Catalan Numbers Associated with a Family of Pascal-like Triangles, J. Int. Seq., Vol. 22 (2019), Article 19.5.8.

Paul Barry, Characterizations of the Borel triangle and Borel polynomials, arXiv:2001.08799 [math.CO], 2020.

P. Barry and A. Hennessy, Four-term Recurrences, Orthogonal Polynomials and Riordan Arrays, Journal of Integer Sequences, 2012, article 12.4.2. - From N. J. A. Sloane, Sep 21 2012

Brian Drake, Limits of areas under lattice paths, Discrete Math. 309 (2009), no. 12, 3936-3953.

M. Dziemianczuk, Counting Lattice Paths With Four Types of Steps, Graphs and Combinatorics, September 2013, Volume 30, Issue 6, pp 1427-1452.

M. Dziemianczuk, On Directed Lattice Paths With Additional Vertical Steps, arXiv preprint arXiv:1410.5747 [math.CO], 2014.

N. J. A. Sloane, Illustration of initial terms of A223092 and A064641

D. V. Kruchinin, On solving some functional equations, Advances in Difference Equations (2015) 2015:17; DOI 10.1186/s13662-014-0347-9.

Index entries for sequences related to boustrophedon transform

Formula

G.f.: (1-x-sqrt(1-6x-3x^2)) / (2x(1+x)). - Brian Drake, Aug 01 2007

G.f.: 1/(1-2x-3x^2/(1-3x-3x^2/(1-3x-3x^2/(1-3x-3x^2/(1-.... (continued fraction). - Paul Barry, Jan 26 2009

a(n) = sum(i=0..n, binomial(n+i,n)*sum(j=0..n+1, binomial(j,-n+2*j-i-2)*binomial(n+1,j)))/(n+1). - Vladimir Kruchinin, May 12 2011

Recurrence: (n+1)*a(n) = (5*n-4)*a(n-1) + 9*(n-1)*a(n-2) + 3*(n-2)*a(n-3). - Vaclav Kotesovec, Oct 13 2012

a(n) ~ 3*(sqrt(6)+sqrt(2))*(3+2*sqrt(3))^n/(8*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 13 2012

G.f.: 1 / (1 - x - (x+x^2) / (1 - x - (x+x^2) / ... )) (continued fraction). - Michael Somos, Mar 30 2014

0 = a(n)*(+9*a(n+1) + 54*a(n+2) + 33*a(n+3) - 12*a(n+4)) + a(n+1)*(+78*a(n+2) + 60*a(n+3) - 27*a(n+4)) + a(n+2)*(+36*a(n+2) + 34*a(n+3) - 14*a(n+4)) + a(n+3)*(+4*a(n+3) + a(n+4)) for all n >= 0. - Michael Somos, Nov 05 2014

D-finite with recurrece (n+1)*a(n) +(-5*n+4)*a(n-1) +9*(-n+1)*a(n-2) +3*(-n+2)*a(n-3)=0. - R. J. Mathar, Oct 16 2017

Example

The array begins
......1
....1...2
..1...5...7
1...8...22..29
G.f. = 1 + 2*x + 7*x^3 + 29*x^4 + 133*x^5 + 650*x^6 + 3319*x^7 + ...

Maple

A:= series( (1-x-sqrt(1-6*x-3*x^2)) / (2*x*(1+x)), x, 21): seq(coeff(A, x, i), i=0..20); # Brian Drake, Aug 01 2007

Mathematica

Table[SeriesCoefficient[(1-x-Sqrt[1-6*x-3*x^2])/(2*x*(1+x)), {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 13 2012 *)

Prog

(PARI) a(n)=if(n<0, 0, polcoeff(serreverse(x*(1-x)/(1+x+x^2)+O(x^(n+2))), n+1)) /* Paul Barry */
(Maxima)
a(n):=sum(binomial(n+i, n)*sum(binomial(j, -n+2*j-i-2)*binomial(n+1, j), j, 0, n+1), i, 0, n)/(n+1); /* Vladimir Kruchinin, May 12 2011 */

Crossrefs

Delannoy numbers: A008288, table: A064642. Cf. A038764, A223092.

Row sums of A201159.

Sequence in context: A368935 A110576 A074600 * A183608 A307389 A104252

Adjacent sequences: A064638 A064639 A064640 * A064642 A064643 A064644

Keyword

nonn

Author

Floor van Lamoen, Oct 03 2001

Status

approved