A201159 -id:A201159 - cp's OEIS frontend

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, 5072534905324615, 30749397292689194

Offset: 0

Floor van Lamoen, Oct 03 2001

nonn

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

			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 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..1000 (terms 0..200 from Vincenzo Librandi)
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

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

Row sums of A201159.

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

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 *)

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 */

a(n)=if(n<0,0,polcoeff(serreverse(x*(1-x)/(1+x+x^2)+O(x^(n+2))),n+1)) /* Paul Barry */

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
a(n) = (-1)^n * (n+1) + Sum_{k=0..n-1} (a(k) + (-1)^k) * (a(n-1-k) + (-1)^(n-1-k)). - Seiichi Manyama, Jul 18 2025

A201075 Irregular triangle read by rows: number of Schroeder paths of length n and weighted area n^2-k.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 3, 4, 3, 3, 3, 1, 1, 1, 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 10, 7, 6, 4, 1, 1, 1, 1, 2, 3, 4, 5, 7, 10, 13, 14, 17, 22, 25, 27, 31, 34, 34, 33, 31, 28, 21, 14, 10, 5, 1, 1, 1, 1, 2, 3, 4, 5, 7, 10, 13, 16, 21, 26, 31, 37, 45, 54

Offset: 0

N. J. A. Sloane, Nov 26 2011

0 <= k <= n^2.

			Triangle begins:
1
1 1
1 1 1 2 1
1 1 1 2 3 4 3 3 3 1
1 1 1 2 3 4 5 7 8 9 10 11 10 7 6 4 1
...

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

Mirror image of A129179.

Cf. A129176, A201076, A201079, A201080, A201159.

gf = Expand /@ FixedPoint[1 + x # (1 + q Normal@# /. {x :> q^2 x}) + O[x]^7 &, 0];
Flatten[Reverse[CoefficientList[#, q]] & /@ CoefficientList[gf, x]] (* Andrey Zabolotskiy, Jan 03 2024 *)

More terms from Andrey Zabolotskiy, Jan 03 2024

A201076 Irregular triangle read by rows: number of {0,2}-shifted Schroeder paths of length n and area k.

Original entry on oeis.org

1, 1, 1, 2, 0, 1, 2, 3, 3, 0, 1, 2, 3, 6, 7, 7, 5, 0, 0, 1, 2, 3, 6, 10, 13, 16, 20, 19, 15, 8, 0, 0, 1, 2, 3, 6, 10, 16, 22, 29, 39, 48, 53, 56, 57, 46, 30, 13, 0, 0, 0, 1, 2, 3, 6, 10, 16, 25, 35, 48, 66, 85, 106, 127, 147, 167, 179, 178, 168, 146, 103, 58, 21, 0, 0, 0

Offset: 0

N. J. A. Sloane, Nov 26 2011

			Triangle begins
1
1
1 2 0
1 2 3 3 0
1 2 3 6 7 7 5 0 0
1 2 3 6 10 13 16 20 19 15 8 0 0
...

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

Row sums give A052709. Rows converge to A101277.

Cf. S-shifted Schroeder paths for various S: A201075 {0,1}, A201079 {0,2,4,6...}, A201080 {0,1,3,5...}, A201159 {0,1,2}.

gf = Expand /@ FixedPoint[1 + (q x + q^2 x^2) # (Normal@# /. {x :> q^2 x}) + O[x]^8 &, 0];
Flatten[Reverse[CoefficientList[#, q]][[;; ;; 2]] & /@ CoefficientList[gf, x]] (* Andrey Zabolotskiy, Jan 02 2024 *)

Row 5 corrected, rows 6-7 added by Andrey Zabolotskiy, Jan 02 2024

A201079 Irregular triangle read by rows: number of {0,2,4,6...}-shifted Schroeder paths of length n and area k.

Original entry on oeis.org

1, 1, 1, 2, 0, 1, 2, 3, 3, 0, 1, 2, 4, 6, 7, 7, 5, 0, 0, 1, 2, 4, 7, 11, 14, 18, 20, 19, 15, 8, 0, 0, 1, 2, 4, 8, 12, 19, 26, 35, 43, 52, 57, 61, 57, 46, 30, 13, 0, 0, 0, 1, 2, 4, 8, 13, 21, 32, 45, 61, 81, 101, 125, 146, 167, 183, 194, 191, 178, 146, 103, 58, 21, 0, 0, 0

Offset: 0

N. J. A. Sloane, Nov 26 2011

			Triangle begins
1
1
1 2 0
1 2 3 3 0
1 2 4 6 7 7 5 0 0
1 2 4 7 11 14 18 20 19 15 8 0 0
1 2 4 8 12 19 26 35 43 52 57 61 57 46 30 13 0 0 0
...

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

Row sums give A063020. Rows converge to A015128.

Cf. S-shifted Schroeder paths for various S: A201075 {0,1}, A201076 {0,2}, A201080 {0,1,3,5...}, A201159 {0,1,2}.

max = 8; s0 = Range[2, max, 2];
gf = Expand /@ FixedPoint[With[{g = Normal@#}, 1 + q x g (g /. {x :> q^2 x}) + Sum[q^(j^2 - j) x^j Product[g /. {x :> q^(2 i - 2) x}, {i, j}], {j, s0}] + O[x]^max] &, 0];
Flatten[Reverse[CoefficientList[#, q]][[;; ;; 2]] & /@ CoefficientList[gf, x]] (* Andrey Zabolotskiy, Jan 02 2024 *)

Name and rows 3 and 5 corrected and row 7 added by Andrey Zabolotskiy, Jan 02 2024

cp's OEIS Frontend

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).

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A201075 Irregular triangle read by rows: number of Schroeder paths of length n and weighted area n^2-k.

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A201076 Irregular triangle read by rows: number of {0,2}-shifted Schroeder paths of length n and area k.

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A201079 Irregular triangle read by rows: number of {0,2,4,6...}-shifted Schroeder paths of length n and area k.

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