A071943 -id:A071943 - cp's OEIS frontend

A108073 Triangle in A071943 with rows reversed.

1, 1, 1, 3, 2, 1, 9, 7, 3, 1, 31, 24, 12, 4, 1, 113, 89, 46, 18, 5, 1, 431, 342, 183, 76, 25, 6, 1, 1697, 1355, 741, 323, 115, 33, 7, 1, 6847, 5492, 3054, 1376, 520, 164, 42, 8, 1, 28161, 22669, 12768, 5900, 2326, 786, 224, 52, 9, 1, 117631, 94962, 54033, 25464

Offset: 0

N. J. A. Sloane, Jun 05 2005

A convolution triangle based on A052709 (with first term omitted). - Philippe Deléham, Sep 15 2005

			1; 1,1; 3,2,1; 9,7,3,1; 31,24,12,4,1; ...

Row sums yield A071356. Column 0 yields A052709.

q:=sqrt(1-4*z-4*z^2): G:=(1-q)/z/(2-t+2*z+t*q): Gserz:=simplify(series(G,z=0,14)): P[0]:=1: for n from 1 to 10 do P[n]:=coeff(Gserz,z^n) od: for n from 0 to 10 do seq(coeff(t*P[n],t^k),k=1..n+1) od; # yields sequence in triangular form # Emeric Deutsch, Jun 06 2005

T[n_, n_] = 1; T[n_, k_] := (k+1)*Sum[Binomial[i, n-k-i] * Binomial[k+2*i, i] / (k+i+1), {i, 1, n-k}]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 29 2015, after Vladimir Kruchinin *)

T(n,k):=if n=k then 1 else k*sum((binomial(i,n-k-i)*binomial(k+2*i-1,i))/(k+i),i,1,n-k); /* Vladimir Kruchinin, Apr 27 2015 */

G.f.: (1-q)/(z(2 - t + 2z + tq)), where q = sqrt(1 - 4z - 4z^2). - Emeric Deutsch, Jun 06 2005
T(0, 0) = 1; T(n, k) = 0 if k < 0 or if k > n; T(n, k) = Sum_{j>=0} T(n-1, k-1+j) + Sum_{j>=0} T(n-1, k+1+j). - Philippe Deléham, Sep 15 2005
T(n,k) = k*Sum_{i=1..(n-k)} C(i,n-k-i)*C(k+2*i-1,i)/(k+i), n > k, T(n,n)=1. - Vladimir Kruchinin, Apr 27 2015

More terms from Emeric Deutsch, Jun 06 2005

A052709 Expansion of g.f. (1-sqrt(1-4x-4x^2))/(2*(1+x)).

Original entry on oeis.org

0, 1, 1, 3, 9, 31, 113, 431, 1697, 6847, 28161, 117631, 497665, 2128127, 9183489, 39940863, 174897665, 770452479, 3411959809, 15181264895, 67833868289, 304256253951, 1369404661761, 6182858317823, 27995941060609, 127100310290431, 578433619525633, 2638370120138751

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

A simple context-free grammar.
Number of lattice paths from (0,0) to (2n-2,0) that stay (weakly) in the first quadrant and such that each step is either U=(1,1), D=(1,-1), or L=(3,1). Equivalently, underdiagonal lattice paths from (0,0) to (n-1,n-1) and such that each step is either (1,0), (0,1), or (2,1). E.g., a(4)=9 because in addition to the five Dyck paths from (0,0) to (6,0) [UDUDUD, UDUUDD, UUDDUD, UUDUDD, UUUDDD] we have LDUD, LUDD, ULDD and UDLD. - Emeric Deutsch, Dec 21 2003
Hankel transform of a(n+1) is A006125(n+1). - Paul Barry, Apr 01 2007
Also, a(n+1) is the number of walks from (0,0) to (n,0) using steps (1,1), (1,-1) and (0,-1). See the U(n,k) array in A071943, where A052709(n+1) = U(n,0). - N. J. A. Sloane, Mar 29 2013
Diagonal sums of triangle in A085880. - Philippe Deléham, Nov 15 2013
From Gus Wiseman, Jun 17 2021: (Start)
Conjecture: For n > 0, also the number of sequences of length n - 1 covering an initial interval of positive integers and avoiding three terms (..., x, ..., y, ..., z, ...) such that x <= y <= z. The version avoiding the strict pattern (1,2,3) is A226316. Sequences covering an initial interval are counted by A000670. The a(1) = 1 through a(4) = 9 sequences are:
  ()  (1)  (1,1)  (1,2,1)
           (1,2)  (1,3,2)
           (2,1)  (2,1,1)
                  (2,1,2)
                  (2,1,3)
                  (2,2,1)
                  (2,3,1)
                  (3,1,2)
                  (3,2,1)
(End)

N. J. A. Sloane, Table of n, a(n) for n = 0..499
Marilena Barnabei, Flavio Bonetti, Niccolò Castronuovo, and Matteo Silimbani, Ascending runs in permutations and valued Dyck paths, Ars Mathematica Contemporanea (2019) Vol. 16, No. 2, 445-463.
Paul Barry, Riordan arrays, generalized Narayana triangles, and series reversion, Linear Algebra and its Applications, 491 (2016) 343-385.
Paul Barry, On a transformation of Riordan moment sequences, arXiv:1802.03443 [math.CO], 2018.
Paul Barry, Characterizations of the Borel triangle and Borel polynomials, arXiv:2001.08799 [math.CO], 2020.
Daniel Birmajer, Juan B. Gil, David S. Kenepp, and Michael D. Weiner, Restricted generating trees for weak orderings, arXiv:2108.04302 [math.CO], 2021.
Daniel Birmajer, Juan B. Gil, Peter R. W. McNamara, and Michael D. Weiner, Enumeration of colored Dyck paths via partial Bell polynomials, arXiv:1602.03550 [math.CO], 2016.
Xiang-Ke Chang, X.-B. Hu, H. Lei, and Y.-N. Yeh, Combinatorial proofs of addition formulas, The Electronic Journal of Combinatorics, 23(1) (2016), #P1.8.
Brian Drake, Limits of areas under lattice paths, Discrete Math. 309 (2009), no. 12, 3936-3953.
M. Dziemianczuk, On Directed Lattice Paths With Additional Vertical Steps, arXiv preprint arXiv:1410.5747 [math.CO], 2014.
James East and Nicholas Ham, Lattice paths and submonoids of Z^2, arXiv:1811.05735 [math.CO], 2018.
L. Ferrari, E. Pergola, R. Pinzani, and S. Rinaldi, Jumping succession rules and their generating functions, Discrete Math., 271 (2003), 29-50.
Nancy S. S. Gu, Nelson Y. Li, and Toufik Mansour, 2-Binary trees: bijections and related issues, Discr. Math., 308 (2008), 1209-1221.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 664
J. P. S. Kung and A. de Mier, Catalan lattice paths with rook, bishop and spider steps, Journal of Combinatorial Theory, Series A 120 (2013) 379-389. - _N. J. A. Sloane_, Dec 27 2012
D. Merlini, D. G. Rogers, R. Sprugnoli, and M. C. Verri, On some alternative characterizations of Riordan arrays, Canad. J. Math., 49 (1997), 301-320.

Diagonal entries of A071943 and A071945.
Cf. A000108, A000670, A025227, A052709, A056986, A071943, A085880.
Cf. A102726, A117434, A158005, A226316, A333217, A335479.

[0] cat [(&+[Binomial(n,k+1)*Binomial(2*k,n-1): k in [0..n-1]])/n: n in [1..30]]; // G. C. Greubel, May 30 2022

spec := [S,{C=Prod(B,Z),S=Union(B,C,Z),B=Prod(S,S)},unlabeled]: seq(combstruct[count](spec,size=n), n=0..20);

InverseSeries[Series[(y-y^2)/(1+y^2), {y, 0, 24}], x] (* then A(x)= y(x) *) (* Len Smiley, Apr 12 2000 *)
CoefficientList[Series[(1 -Sqrt[1 -4x -4x^2])/(2(1+x)), {x, 0, 33}], x] (* Vincenzo Librandi, Feb 12 2016 *)

a(n)=polcoeff((1-sqrt(1-4*x*(1+x+O(x^n))))/2/(1+x),n)

[sum(binomial(k, n-k-1)*catalan_number(k) for k in (0..n-1)) for n in (0..30)] # G. C. Greubel, May 30 2022

a(n) + a(n-1) = A025227(n).
a(n) = Sum_{k=0..floor((n-1)/2)} (2*n-2-2*k)!/(k!*(n-k)!*(n-1-2*k)!). - Emeric Deutsch, Nov 14 2001
D-finite with recurrence: n*a(n) = (3*n-6)*a(n-1) + (8*n-18)*a(n-2) + (4*n-12)*a(n-3), n>2. a(1)=a(2)=1.
a(n) = b(1)*a(n-1) + b(2)*a(n-2) + ... + b(n-1)*a(1) for n>1 where b(n)=A025227(n).
G.f.: A(x) = x/(1-(1+x)*A(x)). - Paul D. Hanna, Aug 16 2002
G.f.: A(x) = x/(1-z/(1-z/(1-z/(...)))) where z=x+x^2 (continued fraction). - Paul D. Hanna, Aug 16 2002; revised by Joerg Arndt, Mar 18 2011
a(n+1) = Sum_{k=0..n} Catalan(k)*binomial(k, n-k). - Paul Barry, Feb 22 2005
From Paul Barry, Mar 14 2006: (Start)
G.f. is x*c(x*(1+x)) where c(x) is the g.f. of A000108.
Row sums of A117434. (End)
a(n+1) = (1/(2*Pi))*Integral_{x=2-2*sqrt(2)..2+2*sqrt(2)} x^n*(4+4x-x^2)/(2*(1+x)). - Paul Barry, Apr 01 2007
From Gary W. Adamson, Jul 22 2011: (Start)
For n>0, a(n) is the upper left term in M^(n-1), where M is an infinite square production matrix as follows:
  1, 1, 0, 0, 0, 0, ...
  2, 1, 1, 0, 0, 0, ...
  2, 2, 1, 1, 0, 0, ...
  2, 2, 2, 1, 1, 0, ...
  2, 2, 2, 2, 1, 1, ...
  ... (End)
G.f.: x*Q(0), where Q(k) = 1 + (4*k+1)*x*(1+x)/(k+1 - x*(1+x)*(2*k+2)*(4*k+3)/(2*x*(1+x)*(4*k+3) + (2*k+3)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 14 2013
a(n) ~ sqrt(2-sqrt(2))*2^(n-1/2)*(1+sqrt(2))^(n-1)/(n^(3/2)*sqrt(Pi)). - Vaclav Kotesovec, Jun 29 2013
a(n+1) = Sum_{k=0..floor(n/2)} A085880(n-k,k). - Philippe Deléham, Nov 15 2013

Better g.f. and recurrence from Michael Somos, Aug 03 2000

More terms from Larry Reeves (larryr(AT)acm.org), Oct 03 2000

A071356 Expansion of (1 - 2x - sqrt(1 - 4x - 4x^2))/(4x^2).

Original entry on oeis.org

1, 2, 6, 20, 72, 272, 1064, 4272, 17504, 72896, 307648, 1312896, 5655808, 24562176, 107419264, 472675072, 2091206144, 9296612352, 41507566592, 186045061120, 836830457856, 3776131489792, 17089399689216, 77548125675520, 352766964908032

Offset: 0

N. J. A. Sloane, Jun 12 2002

nonn

Number of underdiagonal lattice paths from (0,0) to the line x=n, using only steps R=(1,0), V=(0,1) and D=(1,2). Also number of Motzkin paths of length n in which both the "up" and the "level" steps come in two colors. E.g., a(2)=6 because we have RR, RVR, RRV, RD, RVRV and RRVV. - Emeric Deutsch, Dec 21 2003
Inverse binomial transform of little Schroeder numbers 1,3,11,... (A001003 with first term deleted). - David Callan, Feb 07 2004
a(n) is the number of planar trees satisfying: 1) Every internal node has at least two children, 2) Among the children of a node, only the leftmost and the rightmost children can be leaves, 3) The tree has n+1 leaves. For instance, a(3)=6. - Marcelo Aguiar (maguiar(AT)math.tamu.edu), Oct 14 2005
Hankel transform is A006125(n+1)=2^C(n+1,2). - Paul Barry, Jan 08 2008
Equals binomial transform of A025235: (1, 1, 3, 7, 21, 61, 191, ...). - Gary W. Adamson, Sep 03 2010
Conjecturally, the number of sequences (e(1), ..., e(n+1)), 0 <= e(i) < i, such that there is no triple i < j < k with e(i) > e(j) <= e(k). [Martinez and Savage, 2.19] - Eric M. Schmidt, Jul 17 2017
Let s denote West's stack-sorting map, and let Av_n(tau_1, ..., tau_r) denote the set of permutations of [n] that avoid the patterns tau_1, ..., tau_r. It is conjectured that a(n) = |s^{-1}(Av_{n+1}(132, 231))| = |s^{-1}(Av_{n+1}(132, 312))| = |s^{-1}(Av_{n+1}(231, 312))|. Only the last of these equalities is known. - Colin Defant, Sep 16 2018

			a(3) = 20 = sum of top row terms in M^3 = (9 + 7 + 3 + 1).

Fung Lam, Table of n, a(n) for n = 0..1465
Marcelo Aguiar and Walter Moreira, Combinatorics of the free Baxter algebra, arXiv:math/0510169 [math.CO], 2005-2007, see Corollary 3.3.iii.
Paul Barry, Characterizations of the Borel triangle and Borel polynomials, arXiv:2001.08799 [math.CO], 2020.
Miklos Bona, Stack-sorting preimages and 0-1-trees, arXiv:2505.18295 [math.CO], 2025. See p. 3.
Vuong Bui, Bounding Klarner's constant from above using a simple recurrence, arXiv:2412.20143 [math.CO], 2024.
Wenqin Cao, Emma Yu Jin, and Zhicong Lin, Enumeration of inversion sequences avoiding triples of relations, Discrete Applied Mathematics (2019); see also author's copy
Colin Defant, Stack-sorting preimages of permutation classes, arXiv:1809.03123 [math.CO], 2018.
Colin Defant, Enumeration of Stack-Sorting Preimages via a Decomposition Lemma, arXiv:1904.02829 [math.CO], 2019.
Serkan Demiriz, Adem Şahin, and Sezer Erdem, Some topological and geometric properties of novel generalized Motzkin sequence spaces, Rendiconti Circ. Mat. Palermo Ser. 2 (2025) Vol. 74, No. 136. See p. 4.
Ivan Geffner and Marc Noy, Counting Outerplanar Maps, Electronic Journal of Combinatorics 24(2) (2017), #P2.3.
Li Guo and Yunnan Li, Braided dendriform and tridendriform algebras and braided Hopf algebras of planar trees, arXiv:1906.06454 [math.QA], 2019.
Bin Han, The gamma-positive coefficients arising in segmented permutations, Discrete Math., 344 (2012), #112336. See p. 7.
Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.
Chetak Hossain, Quotients Derived from Posets in Algebraic and Topological Combinatorics, Ph. D. Dissertation, North Carolina State University (2019).
Germain Kreweras, Sur les hiérarchies de segments, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, #20 (1973), p. 32-33 (same sequence but with offset 1).
Germain Kreweras, Sur les hiérarchies de segments, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, #20 (1973). (Annotated scanned copy)
Megan A. Martinez and Carla D. Savage, Patterns in Inversion Sequences II: Inversion Sequences Avoiding Triples of Relations, arXiv:1609.08106 [math.CO], 2016.
Donatella Merlini, Douglas G. Rogers, Renzo Sprugnoli, and M. Cecilia Verri, On some alternative characterizations of Riordan arrays, Canad. J. Math., 49 (1997), 301-320.
Louis W. Shapiro and Carol J. Wang, A bijection between 3-Motzkin paths and Schroder paths with no peak at odd height, JIS 12 (2009) 09.3.2.

A036774(n) = a(n-1) * n! / 2^(n-1).
Row sums of A071943.
Cf. A000108, A025235.

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!((1 - 2*x - Sqrt(1 - 4*x - 4*x^2))/(4*x^2))); // Vincenzo Librandi, Jan 21 2020

CoefficientList[Series[(1-2*x-Sqrt[1-4*x-4*x^2])/(4*x^2), {x, 0, 20}], x] (* Vaclav Kotesovec, Sep 24 2013 *)
a[n_] := 2^n Hypergeometric2F1[(1-n)/2, -n/2, 2, 2];
Table[a[n], {n, 0, 24}] (* Peter Luschny, May 30 2021 *)

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

{a(n)= if(n<1, n==0, polcoeff( 2/(1 -2*x +sqrt(1 -4*x -4*x^2 +x*O(x^n))), n))}

{a(n)= local(A); if(n<0, 0, A= x*O(x^n); n!*simplify(polcoeff( exp(2*x +A)* besseli(1, 2*x* quadgen(8) +A), n)))} /* Michael Somos, Mar 31 2007 */

def A071356_list(n):  # n>=1
    T = [0]*(n+1); R = [1]
    for m in (1..n-1):
        a,b,c = 1,0,0
        for k in range(m,-1,-1):
            r = a + 2*(b + c)
            if k < m : T[k+2] = u;
            a,b,c = T[k-1],a,b
            u = r
        T[1] = u; R.append(u)
    return R
A071356_list(25)  # Peter Luschny, Nov 01 2012

G.f. A(x) satisfies 2x^2*A(x)^2+(2x-1)*A(x)+1=0 and A(x)=1/(1-2x-2x^2/A(x)). - Michael Somos, Sep 06 2003
a(n) = Sum_{k=0..floor(n/2)} C(n, 2k)C(k)2^(n-2k)*2^k. - Paul Barry, May 18 2005
G.f.: (1 - 2*x - sqrt(1 - 4*x - 4*x^2) )/(4*x^2) = 2/(1 - 2*x +sqrt(1 - 4*x - 4*x^2)).
Moment representation is a(n) = (1/(4*Pi))*int(x^n*sqrt(4-4x-x^2), x, -2*sqrt(2)-2, 2*sqrt(2)-2). - Paul Barry, Jan 08 2008
G.f.: 1/(1-2x-2x^2/(1-2x-2x^2/(1-2x-2x^2/(1-2x-2x^2/(1-2x-2x^2/.... (continued fraction). - Paul Barry, Dec 06 2008
From Gary W. Adamson, Jul 22 2011: (Start)
a(n) = sum of top row terms of M^n, M = an infinite square production matrix as follows:
  1, 1, 0, 0, 0, 0, ...
  2, 1, 1, 0, 0, 0, ...
  2, 2, 1, 1, 0, 0, ...
  2, 2, 2, 1, 1, 0, ...
  2, 2, 2, 2, 1, 1, ...
  2, 2, 2, 2, 2, 1, ... (End)
E.g.f.: a(n) = n!* [x^n] exp(2*x)*BesselI(1, 2*sqrt(2)*x)/(sqrt(2)*x). - Peter Luschny, Aug 25 2012
D-finite with recurrence: (n+2)*a(n) +2*(-2*n-1)*a(n-1) +4*(-n+1)*a(n-2)=0. - R. J. Mathar, Dec 02 2012 (Formula verified and used for computations. - Fung Lam, Feb 24 2014)
a(n) ~ 2^(n - 1/4) * (1+sqrt(2))^(n + 3/2) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Sep 24 2013, simplified Jan 26 2019
a(n) = A179190(n+2)/4. - R. J. Mathar, Jan 20 2020
a(n) = 2^n * hypergeom((1 - n)/2, -n/2, 2, 2). - Peter Luschny, May 30 2021
a(n) = (-2*î)^(n+2) * (Legendre_P(n+2, i) - Legendre_P(n, i))/(4*(2*n + 3)). - Peter Bala, May 06 2024
From Emanuele Munarini, Jun 13 2024: (Start)
a(n) = Sum_{k=0..floor(n/2)} binomial(n, k)*binomial(n-k, k)*2^(n-k)/(k+1).
a(n) = Sum_{k=0..floor((n+2)/3)} binomial(n-2k+2, 2k)*Catalan(n-2k+1).
a(n) = Sum_{k=0..floor((n+2)/4)} binomial(n-2k+1, 2k+1)*Catalan(n-2k). (End)

A071946 Triangle T(n,k) read by rows giving number of underdiagonal lattice paths from (0,0) to (n,k) using only steps R = (1,0), V = (0,1) and D = (3,1).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 4, 6, 6, 1, 6, 13, 19, 19, 1, 8, 23, 44, 63, 63, 1, 10, 37, 87, 156, 219, 219, 1, 12, 55, 155, 330, 568, 787, 787, 1, 14, 77, 255, 629, 1260, 2110, 2897, 2897, 1, 16, 103, 395, 1111, 2527, 4856, 7972, 10869, 10869, 1, 18, 133, 583, 1849, 4706, 10130, 18889, 30545, 41414, 41414

Offset: 0

N. J. A. Sloane, Jun 15 2002

			Triangle T(n,k) begins:
  1;
  1, 1;
  1, 2,  2;
  1, 4,  6,  6;
  1, 6, 13, 19, 19;
  ...

Alois P. Heinz, Rows n = 0..150, flattened
D. Merlini, D. G. Rogers, R. Sprugnoli and M. C. Verri, On some alternative characterizations of Riordan arrays, Canad J. Math., 49 (1997), 301-320.

Related arrays: A071943, A071944, A071945.

A108076 is the reverse, A119254 is the row sums and A071969 is the last (largest) number in each row.

T:= proc(n, k) option remember; `if`(n=0 and k=0, 1,
     `if`(k<0 or nAlois P. Heinz, May 05 2023

T[n_, k_] := T[n, k] = If[n == 0 && k == 0, 1,
   If[k < 0 || n < k, 0, T[n-1, k] + T[n, k-1] + T[n-3, k-1]]];
Table[Table[T[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Jan 25 2025, after Alois P. Heinz *)

More terms from Joshua Zucker, May 10 2006

A378323 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,0) = 0^n and T(n,k) = k * Sum_{r=0..n} binomial(3r+k,r) * binomial(r,n-r)/(3*r+k) for k > 0.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 4, 0, 1, 3, 9, 18, 0, 1, 4, 15, 44, 94, 0, 1, 5, 22, 79, 240, 529, 0, 1, 6, 30, 124, 450, 1390, 3135, 0, 1, 7, 39, 180, 737, 2685, 8404, 19270, 0, 1, 8, 49, 248, 1115, 4532, 16585, 52426, 121732, 0, 1, 9, 60, 329, 1599, 7066, 28624, 105147, 334964, 785496, 0

Offset: 0

Seiichi Manyama, Nov 23 2024

			Square array begins:
  1,    1,    1,     1,     1,     1,     1, ...
  0,    1,    2,     3,     4,     5,     6, ...
  0,    4,    9,    15,    22,    30,    39, ...
  0,   18,   44,    79,   124,   180,   248, ...
  0,   94,  240,   450,   737,  1115,  1599, ...
  0,  529, 1390,  2685,  4532,  7066, 10440, ...
  0, 3135, 8404, 16585, 28624, 45655, 69021, ...

Columns k=0..2 give A000007, A364475, A371576.
Cf. A038137, A071943.
Cf. A378318.

T(n, k, t=3, u=0) = if(k==0, 0^n, k*sum(r=0, n, binomial(t*r+u*(n-r)+k, r)*binomial(r, n-r)/(t*r+u*(n-r)+k)));
matrix(7, 7, n, k, T(n-1, k-1))

G.f. A_k(x) of column k satisfies A_k(x) = ( 1 + x * (1 + x) * A_k(x)^(3/k) )^k for k > 0.
G.f. of column k: B(x)^k where B(x) is the g.f. of A364475.
B(x)^k = B(x)^(k-1) + x * B(x)^(k+2) + x^2 * B(x)^(k+2). So T(n,k) = T(n,k-1) + T(n-1,k+2) + T(n-2,k+2) for n > 1.

A223092 Triangle read by rows: let T(n,k) (for n >= 0, 0 <= k <= n) be the number of walks from (0,0) to (n,k) using steps (1,1), (1,0), (1,-1) and (0,-1); n-th row of triangle gives T(n,n), T(n,n-1), ..., T(n,0).

Original entry on oeis.org

1, 1, 2, 1, 4, 7, 1, 6, 18, 29, 1, 8, 33, 86, 133, 1, 10, 52, 179, 431, 650, 1, 12, 75, 316, 978, 2238, 3319, 1, 14, 102, 505, 1874, 5406, 11941, 17498, 1, 16, 133, 754, 3235, 11020, 30241, 65086, 94525, 1, 18, 168, 1071, 5193, 20202, 64698, 171045, 360897, 520508, 1, 20, 207, 1464, 7896, 34362, 124455, 380400, 977040, 2029490, 2910895

Offset: 0

N. J. A. Sloane, Mar 29 2013

			Triangle begins:
[1]
[1, 2]
[1, 4, 7]
[1, 6, 18, 29]
[1, 8, 33, 86, 133]
[1, 10, 52, 179, 431, 650]
[1, 12, 75, 316, 978, 2238, 3319]
...
The T(n,k) array begins:
4:  0  0  0  0   1  10 ...
3:  0  0  0  1   8  52 ...
2:  0  0  1  6  33 179 ...
1:  0  1  4 18  86 431 ...
0:  1  2  7 29 133 650 ...
-------------------------
k/n:0  1  2  3   4   5 ...
T(5,2) = T(5,3) + T(4,3) + T(4,2) + T(4,1) = 52 + 8 + 33 + 86 = 179.- _Philippe Deléham_, Mar 29 2013
This is also Dziemianczuk's array N(-i,i+j) read by antidiagonals:
1,2,7,29,133,650,3319,17498, ...
1,4,18,86,431,2238,11941,65086, ...
1,6,33,179,978,5406,30241,171045, ...
1,8,52,316,1874,11020,64698,380400, ...
1,10,75,505,3235,20202,124455,761160, ...
... - _N. J. A. Sloane_, Dec 05 2013

Alois P. Heinz, Rows n = 0..140, flattened
M. Dziemianczuk, Counting Lattice Paths With Four Types of Steps, Graphs and Combinatorics, September 2013
N. J. A. Sloane, Illustration of initial terms of A223092 and A064641

Cf. A064641 (T(n,0)), A071943, A052709.

T:= proc(n, k) option remember; `if`(n=0 and k=0, 1,
      `if`(n<0 or k<0 or k>n, 0, add(T(n-l[1], k-l[2]),
       l=[[1, 1], [1, 0], [1, -1], [0, -1]]) ))
    end:
seq(seq(T(n, n-j), j=0..n), n=0..10);  # Alois P. Heinz, Apr 08 2013

max = 10; T[0, 0] = 1; T[n_ /; n >= 0, k_ /; 0 <= k <= max] := T[n, k] = T[n, k+1]+T[n-1, k+1]+T[n-1, k]+T[n-1, k-1]; T[n_, k_] = 0; Table[Table[T[n, k], {k, n, 0, -1}], {n, 0, max}] // Flatten (* Jean-François Alcover, Mar 07 2014, after Philippe Deléham *)

T(n,k) = T(n,k+1) + T(n-1,k+1) + T(n-1,k) + T(n-1,k-1). - Philippe Deléham, Mar 29 2013

cp's OEIS Frontend

A108073 Triangle in A071943 with rows reversed.

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A052709 Expansion of g.f. (1-sqrt(1-4*x-4*x^2))/(2*(1+x)).

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A071356 Expansion of (1 - 2*x - sqrt(1 - 4*x - 4*x^2))/(4*x^2).

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A071946 Triangle T(n,k) read by rows giving number of underdiagonal lattice paths from (0,0) to (n,k) using only steps R = (1,0), V = (0,1) and D = (3,1).

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A378323 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,0) = 0^n and T(n,k) = k * Sum_{r=0..n} binomial(3r+k,r) * binomial(r,n-r)/(3*r+k) for k > 0.

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A223092 Triangle read by rows: let T(n,k) (for n >= 0, 0 <= k <= n) be the number of walks from (0,0) to (n,k) using steps (1,1), (1,0), (1,-1) and (0,-1); n-th row of triangle gives T(n,n), T(n,n-1), ..., T(n,0).

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A052709 Expansion of g.f. (1-sqrt(1-4x-4x^2))/(2*(1+x)).

A071356 Expansion of (1 - 2x - sqrt(1 - 4x - 4x^2))/(4x^2).