A000958 -id:A000958 - cp's OEIS frontend

A127539 Number of ordered trees with n edges having no odd-length branches starting at the root.

1, 0, 1, 0, 3, 3, 16, 37, 134, 411, 1411, 4747, 16500, 57671, 204380, 730032, 2629637, 9535268, 34787215, 127585608, 470162614, 1739952061, 6463845941, 24096378885, 90112499714, 337965831635, 1270901550454, 4790836498608, 18100497143361

Offset: 0

Emeric Deutsch, Mar 01 2007

nonn

a(n)=A127538(n,0).

			a(3)=0 because all five ordered trees with 3 edges have at least one odd-length branch starting at the root.

Cf. A127538, A000958.

C:=(1-sqrt(1-4*z))/2/z: G:=(1+z)*C/(C+z): Gser:=series(G,z=0,35): seq(coeff(Gser,z,n),n=0..31);

G.f.=(1+z)*C/(C+z), where C =[1-sqrt(1-4z)]/(2z) is the Catalan function.

D-finite with recurrence (-n+1)*a(n) +2*(n-3)*a(n-1) +(7*n-25)*a(n-2) +(3*n-17)*a(n-3) +(3*n-7)*a(n-4) +2*(2*n-9)*a(n-5)=0. - R. J. Mathar, Jul 26 2022

A127541 Triangle read by rows: T(n,k) is the number of ordered trees with n edges having k even-length branches starting at the root (0<=k<=n).

Original entry on oeis.org

1, 1, 1, 1, 3, 2, 8, 5, 1, 24, 15, 3, 75, 46, 10, 1, 243, 148, 34, 4, 808, 489, 116, 16, 1, 2742, 1652, 402, 61, 5, 9458, 5678, 1408, 228, 23, 1, 33062, 19792, 4982, 847, 97, 6, 116868, 69798, 17783, 3138, 393, 31, 1, 417022, 248577, 63967, 11627, 1557, 143, 7

Offset: 0

Emeric Deutsch, Mar 01 2007

Row n has 1+floor(n/2) terms. Row sums are the Catalan numbers (A000108). T(n,0)=A000958(n-1). Sum(k*T(n,k),k=0..floor(n/2))=A127540(n-1).

			T(2,0)=1 because we have the tree /\.
Triangle starts:
1;
1;
1,1;
3,2;
8,5,1;
24,15,3;

Cf. A000108, A000958, A127538, A127540.

C:=(1-sqrt(1-4*z))/2/z: G:=(1+z)/(1+z-z*C-t*z^2*C): Gser:=simplify(series(G,z=0,17)): for n from 0 to 14 do P[n]:=sort(coeff(Gser,z,n)) od: for n from 0 to 14 do seq(coeff(P[n],t,j),j=0..floor(n/2)) od; # yields sequence in triangular form

G.f.=(1+z)/(1+z-z*C-tz^2*C), where C =[1-sqrt(1-4z)]/(2z) is the Catalan function.

A167769 Pendular trinomial triangle (p=0), read by rows of 2n+1 terms (n>=0), defined by the recurrence : if 0

Original entry on oeis.org

1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 2, 3, 2, 1, 0, 0, 1, 3, 6, 8, 6, 3, 1, 0, 0, 1, 4, 10, 18, 24, 18, 10, 4, 1, 0, 0, 1, 5, 15, 33, 57, 75, 57, 33, 15, 5, 1, 0, 0, 1, 6, 21, 54, 111, 186, 243, 186, 111, 54, 21, 6, 1, 0, 0, 1, 7, 28, 82, 193, 379, 622, 808, 622, 379, 193, 82, 28, 7, 1, 0, 0

Offset: 0

Philippe Deléham, Nov 11 2009

See A119369 for p=1 and A122445 for p=2. The diagonals may be generated by iterated convolutions of a base sequence B (A000108(n)) with the sequence C (A000957(n+1)) of central terms.

			Triangle begins :
  1;
  1, 0,  0;
  1, 1,  1,  0,  0;
  1, 2,  3,  2,  1,  0,  0;
  1, 3,  6,  8,  6,  3,  1,  0,  0;
  1, 4, 10, 18, 24, 18, 10,  4,  1, 0, 0,
  1, 5, 15, 33, 57, 75, 57, 33, 15, 5, 1, 0, 0; ...

Kim, Ki Hang; Rogers, Douglas G.; Roush, Fred W. Similarity relations and semiorders. Proceedings of the Tenth Southeastern Conference on Combinatorics, Graph Theory and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1979), pp. 577--594, Congress. Numer., XXIII-XXIV, Utilitas Math., Winnipeg, Man., 1979. MR0561081 (81i:05013) - From N. J. A. Sloane, Jun 05 2012

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A000108, A000957, A000958, A001558, A001559, A071724, A104629.

T:= proc(n, k) option remember;
      if k=0 and n=0 then 1;
    elif k<0 or k>2*(n-1) then 0;
    elif n=2 and k<3 then 1;
    elif kG. C. Greubel, Mar 17 2021

T[n_, k_]:= T[n, k]= If[k==0 && n==0, 1, If[k<0 || k>2*(n-1), 0, If[n==2 && k<3, 1, If[kG. C. Greubel, Mar 17 2021 *)

T(n, k)=if(k==0 && n==0, 1, if(k>2*n-2 || k<0, 0, if(n==2 && k<=2, 1, if(kPaul D. Hanna, Nov 12 2009

@CachedFunction
def T(n, k):
    if (k==0 and n==0): return 1
    elif (k<0 or k>2*(n-1)): return 0
    elif (n==2 and k<3): return 1
    elif (kG. C. Greubel, Mar 17 2021

Sum_{k=0..2*n} T(n,k) = A071724(n) = [n=0] + 3*binomial(2n,n-1)/(n+2) = [n=0] + n*C(n)/(n+2), where C(n) are the Catalan numbers (A000108). - G. C. Greubel, Mar 17 2021

A193215 Number of Dyck paths of semilength n having the property that the heights of the first and the last peaks coincide.

Original entry on oeis.org

1, 2, 3, 6, 14, 38, 113, 356, 1164, 3906, 13364, 46426, 163294, 580316, 2080475, 7515038, 27324014, 99920756, 367264130, 1356043388, 5027345564, 18706888196, 69841532210, 261545298848, 982175296016, 3697785571820, 13954630170720, 52776659865348, 200006396351216, 759386612309146, 2888310863702017

Offset: 1

Volodymyr Mazorchuk, Aug 26 2011

a(n+1) - a(n) = A000958(n) (this reduces to David Callan's comment on A000958(n) from Aug 23 2011).

The sequence gives the trace of the matrix describing the statistics of Dyck paths of semilength n with respect to the heights of the first and the last peaks, see the paper of Baur and Mazorchuk.

G. C. Greubel, Table of n, a(n) for n = 1..1000
K. Baur and V. Mazorchuk, Combinatorial analogues of ad-nilpotent ideals for untwisted affine Lie algebras, arXiv:1108.3659 [math.RA], 2011.

for n from 1 by 1 to 100 do 1+sum(binomial(2*n-2-2*k, n-1-k)-binomial(2*n-2-2*k, n-1-2*k), k = 1 .. n-1) end do

Table[1+Sum[Binomial[2*n-2-2*k, n-1-k]-Binomial[2*n-2-2*k, n-1-2*k],{k,1,n-1}],{n,1,20}] (* Vaclav Kotesovec, Mar 21 2014 *)

a(n)=1+sum(k=1,n-1,binomial(2*n-2-2*k, n-1-k)-binomial(2*n-2-2*k, n-1-2*k));

a(n) = 1 + Sum_{i=1..n-1} A000958(i).
Recurrence: 2*n*(5*n-11)*a(n) = 3*(15*n^2 - 53*n + 40)*a(n-1) - 3*(5*n^2 - 21*n + 20)*a(n-2) - 2*(2*n-5)*(5*n-6)*a(n-3). - Vaclav Kotesovec, Mar 21 2014
a(n) ~ 5*4^n/(27*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 21 2014

A236918 Triangle read by rows: Catalan triangle of the k-Fibonacci sequence.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 1, 3, 7, 8, 1, 4, 12, 22, 24, 1, 5, 18, 43, 73, 75, 1, 6, 25, 72, 156, 246, 243, 1, 7, 33, 110, 283, 564, 844, 808, 1, 8, 42, 158, 465, 1092, 2046, 2936, 2742, 1, 9, 52, 217, 714, 1906, 4178, 7449, 10334, 9458, 1, 10, 63, 288, 1043, 3096, 7670, 15904, 27231, 36736, 33062

Offset: 1

N. J. A. Sloane, Feb 09 2014

Reversal of the Riordan array (f(x), x*f(x)) where f(x) is the g.f. of A000958. - Philippe Deléham, Feb 10 2014

Row sums are in A109262. - Philippe Deléham, Feb 10 2014

			Triangle begins:
  1;
  1,  1;
  1,  2,  3;
  1,  3,  7,   8;
  1,  4, 12,  22,   24;
  1,  5, 18,  43,   73,   75;
  1,  6, 25,  72,  156,  246,  243;
  1,  7, 33, 110,  283,  564,  844,   808;
  1,  8, 42, 158,  465, 1092, 2046,  2936,  2742;
  1,  9, 52, 217,  714, 1906, 4178,  7449, 10334,  9458;
  1, 10, 63, 288, 1043, 3096, 7670, 15904, 27231, 36736, 33062;
  ... - Extended by _Philippe Deléham_, Feb 10 2014

G. C. Greubel, Rows n = 1..50 of the triangle, flattened
Sergio Falcon, Catalan transform of the K-Fibonacci sequence, Commun. Korean Math. Soc. 28 (2013), No. 4, pp. 827-832.

Diagonals give A000958, A114495.

Cf. A109262 (row sums).

P[n_, x_]:= P[n,x]= If[n==0, 1, Sum[(j/(2*n-j))*Binomial[2*n-j, n-j]*Fibonacci[j, 1/x] *x^(n-1), {j,0,n}]];
T[n_, k_]:= Coefficient[P[n, x], x, k];
Table[T[n, k], {n,10}, {k, 0, n-1}]//Flatten (* G. C. Greubel, Jun 14 2022 *)

def f(n,x): return sum( binomial(n-j-1, j)*x^(n-2*j-1) for j in (0..(n-1)//2) )
def p(n,x):
    if (n==0): return 1
    else: return sum( (j/(2*n-j))*binomial(2*n-j, n-j)*x^(n-1)*f(j, 1/x) for j in (0..n) )
def A236918(n,k): return ( p(n,x) ).series(x, n+1).list()[k]
flatten([[A236918(n,k) for k in (0..n-1)] for n in (1..12)]) # G. C. Greubel, Jun 14 2022

T(n, k) = coefficient of [x^k]( p(n, x) ), where p(n, x) = Sum_{j=0..n} (j/(2*n-j))*binomial(2*n-j, n-j)*x^(n-1)*Fibonacci(j, 1/x) with p(0, x) = 1 and Fibonacci(n, x) are the Fibonacci polynomials. - G. C. Greubel, Jun 14 2022

More terms from Philippe Deléham, Feb 09 2014

A237619 Riordan array (1/(1+xc(x)), xc(x)) where c(x) is the g.f. of Catalan numbers (A000108).

Original entry on oeis.org

1, -1, 1, 0, 0, 1, -1, 1, 1, 1, -2, 2, 3, 2, 1, -6, 6, 8, 6, 3, 1, -18, 18, 24, 18, 10, 4, 1, -57, 57, 75, 57, 33, 15, 5, 1, -186, 186, 243, 186, 111, 54, 21, 6, 1, -622, 622, 808, 622, 379, 193, 82, 28, 7, 1, -2120, 2120, 2742, 2120, 1312, 690, 311, 118, 36, 8, 1

Offset: 0

Philippe Deléham, Feb 10 2014

			Triangle begins:
    1;
   -1,  1;
    0,  0,  1;
   -1,  1,  1,  1;
   -2,  2,  3,  2,  1;
   -6,  6,  8,  6,  3,  1;
  -18, 18, 24, 18, 10,  4, 1;
  -57, 57, 75, 57, 33, 15, 5, 1;
Production matrix begins:
  -1, 1;
  -1, 1, 1;
  -1, 1, 1, 1;
  -1, 1, 1, 1, 1;
  -1, 1, 1, 1, 1, 1;
  -1, 1, 1, 1, 1, 1, 1;
  -1, 1, 1, 1, 1, 1, 1, 1;
  -1, 1, 1, 1, 1, 1, 1, 1, 1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Diagonals: A000012, A000217, A023443, A166830.
Cf. A000957, A000958, A001558, A001559, A104629, A126983.
Cf. A065602, A167772.

A065602[n_, k_]:= A065602[n, k]= Sum[(k-1+2*j)*Binomial[2*(n-j)-k-1, n-1]/(2*(n - j) -k-1), {j,0,(n-k)/2}];
T[n_, k_]:= If[k==0, A065602[n, 0], If[n==1 && k==1, 1, A065602[n, k]]];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, May 27 2022 *)

def A065602(n, k): return sum( (k+2*j-1)*binomial(2*n-2*j-k-1, n-1)/(2*n-2*j-k-1) for j in (0..(n-k)//2) )
def A237619(n, k):
    if (n<2): return (-1)^(n-k)
    elif (k==0): return A065602(n, 0)
    else: return A065602(n, k)
flatten([[A237619(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 27 2022

Sum_{k=0..n} T(n,k)*x^k = A126983(n), A000957(n+1), A026641(n) for x = 0, 1, 2 respectively.
T(n, k) = A167772(n-1, k-1) for k > 0, with T(n, 0) = A167772(n, 0).
T(n, 0) = A126983(n).
T(n+1, 1) = A000957(n+1).
T(n+2, 2) = A000958(n+1).
T(n+3, 3) = A104629(n) = A000957(n+3).
T(n+4, 4) = A001558(n).
T(n+5, 5) = A001559(n).
T(n, k) = A065602(n, k) for k > 0, with T(n, k) = (-1)^(n-k), for n < 2, and T(n, 0) = A065602(n, 0). - G. C. Greubel, May 27 2022

A098747 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n having exactly k UDU's at low level.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 8, 4, 1, 1, 24, 11, 5, 1, 1, 75, 35, 14, 6, 1, 1, 243, 113, 47, 17, 7, 1, 1, 808, 376, 156, 60, 20, 8, 1, 1, 2742, 1276, 532, 204, 74, 23, 9, 1, 1, 9458, 4402, 1840, 712, 257, 89, 26, 10, 1, 1, 33062, 15390, 6448, 2507, 917, 315, 105, 29, 11, 1, 1, 116868

Offset: 1

N. J. A. Sloane, Oct 30 2004

T(n,0) = A000958(n-1). - Emeric Deutsch, Dec 23 2006

			Triangle begins:
1
1 1
3 1 1
8 4 1 1
24 11 5 1 1
75 35 14 6 1 1
T(4,2)=1 because we have UDUDUUDD.

Yidong Sun, The statistic "number of udu's" in Dyck paths, Discrete Math., 287 (2004), 177-186.

Cf. A091869, A092107.

Cf. A000958.

c:=(1-sqrt(1-4*z))/2/z: G:=z*c/(1-t*z+z-z*c): Gser:=simplify(series(G,z=0,15)): for n from 1 to 13 do P[n]:=sort(coeff(Gser,z,n)) od: for n from 1 to 12 do seq(coeff(P[n],t,k),k=0..n-1) od; # yields sequence in triangular form - Emeric Deutsch, Dec 23 2006

u[n_, k_, i_]:=(2i+1)/(n-k)Binomial[k+i, i]Binomial[2n-2k-2i-2, n-k-1] u[n_, k_]/;k<=n-1 := Sum[u[n, k, i], {i, 0, n-k-1}] Table[u[n, k], {n, 10}, {k, 0, n-1}] (* u[n, k, i] is the number of Dyck n-paths with k low UDUs and k+i+1 returns altogether. For example, with n=4, k=1 and i=1, u[n, k, i] counts UDUUDDUD, UUDDUDUD because each has size n=4, k=1 low UDUs and k+i+1=3 returns to ground level. *) (* David Callan, Nov 03 2005 *)

See Mathematica line.
G.f.=zC/(1+z-tz-zC), where C=(1-sqrt(1-4z))/(2z) is the Catalan function. - Emeric Deutsch, Dec 23 2006
With offset 0 (0<=k<=n), T(n,k)=A065600(n,k)+A065600(n+1,k)-A065600(n,k-1). - Philippe Deléham, Apr 01 2007

A187913 Generalized Riordan array based on the Fine's numbers A000957.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 2, 4, 1, 1, 6, 10, 5, 2, 1, 18, 32, 13, 9, 2, 1, 57, 100, 44, 28, 10, 3, 1, 186, 329, 142, 100, 32, 15, 3, 1, 622, 1101, 480, 344, 119, 55, 16, 4, 1, 2120, 3761, 1640, 1214, 420, 216, 60, 22, 4, 1, 7338, 13035, 5698, 4300, 1517, 810, 243, 92, 23, 5, 1

Offset: 0

Paul Barry, Mar 15 2011

First column is the Fine's numbers A000957. Row sums are A000958. Inverse binomial transform of A187914.

			Triangle begins
1,
0, 1,
1, 1, 1,
2, 4, 1, 1,
6, 10, 5, 2, 1,
18, 32, 13, 9, 2, 1,
57, 100, 44, 28, 10, 3, 1,
186, 329, 142, 100, 32, 15, 3, 1,
622, 1101, 480, 344, 119, 55, 16, 4, 1,
2120, 3761, 1640, 1214, 420, 216, 60, 22, 4, 1,
7338, 13035, 5698, 4300, 1517, 810, 243, 92, 23, 5, 1
Production matrix is
0, 1,
1, 1, 1,
1, 2, 0, 1,
1, 2, 1, 1, 1,
1, 2, 1, 2, 0, 1,
1, 2, 1, 2, 1, 1, 1,
1, 2, 1, 2, 1, 2, 0, 1,
1, 2, 1, 2, 1, 2, 1, 1, 1,
1, 2, 1, 2, 1, 2, 1, 2, 0, 1
Thus
57=1.0+0.18+1.32+1.13+1.9+1.2+1.1;
100=1.18+1.32+2.13+2.9+2.2+2.1;
44=1.32+0.13+1.9+1.2+1.1

Let g(x)=(1+2x-sqrt(1-4x))/(2x(2+x)) be the g.f. of the Fine's numbers A000957. Then column k has

g.f. x^k*g(x)^(k+1)/(1-xg(x)-x^2g(x)^2)^floor((k+1)/2).

A323212 The Fibonacci-Catalan Hybrid. Expansion of 1 + x(2(x + 1))/(sqrt(1 - 4y) - 2x*(x + 1) + 1). Square array read by descending antidiagonals, A(n,k) for n,k >= 0.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 2, 3, 3, 0, 5, 7, 7, 5, 0, 14, 19, 19, 15, 8, 0, 42, 56, 56, 46, 30, 13, 0, 132, 174, 174, 146, 103, 58, 21, 0, 429, 561, 561, 477, 351, 220, 109, 34, 0, 1430, 1859, 1859, 1595, 1205, 801, 453, 201, 55, 0, 4862, 6292, 6292, 5434, 4180, 2884, 1756, 908, 365, 89

Offset: 0

Peter Luschny, Feb 14 2019

			      1,   0,    0,    0,     0,      0,      0,       0,       0, ...
      1,   1,    2,    5,    14,     42,    132,     429,    1430, ... [A000108]
      2,   3,    7,   19,    56,    174,    561,    1859,    6292, ... [A005807]
      3,   7,   19,   56,   174,    561,   1859,    6292,   21658, ... [A005807]
      5,  15,   46,  146,   477,   1595,   5434,   18798,   65858, ...
      8,  30,  103,  351,  1205,   4180,  14651,   51844,  185028, ...
     13,  58,  220,  801,  2884,  10372,  37401,  135420,  492558, ...
     21, 109,  453, 1756,  6621,  24674,  91532,  339184, 1257762, ...
     34, 201,  908, 3734, 14719,  56796, 216698,  821848, 3107583, ...
     55, 365, 1781, 7746, 31872, 127245, 499164, 1937439, 7470819, ...
A000045,A023610,...
Seen as a triangle a refinement of A000958:
[0]                                1
[1]                              0, 1
[2]                            0, 1, 2
[3]                           0, 2, 3, 3
[4]                         0, 5, 7, 7, 5
[5]                      0, 14, 19, 19, 15, 8
[6]                   0, 42, 56, 56, 46, 30, 13
[7]               0, 132, 174, 174, 146, 103, 58, 21
[8]            0, 429, 561, 561, 477, 351, 220, 109, 34
[9]       0, 1430, 1859, 1859, 1595, 1205, 801, 453, 201, 55

Antidiagonal sums (or row sums of the triangle) are A000958.

Cf. A000045, A000108, A023610, A005807.

gf := 1 + x*(2*(x + 1))/(sqrt(1 - 4*y) - 2*x*(x + 1) + 1):
serx := series(gf, x, 20): sery := n -> series(coeff(serx, x, n), y, 20):
row := n -> seq(coeff(sery(n), y, j), j=0..9):
seq(lprint(row(n)), n=0..9);

m = 11; T = PadRight[CoefficientList[#+O[y]^m, y], m]& /@ CoefficientList[1 + 2x(x+1)/(Sqrt[1-4y] - 2x(x+1) + 1) + O[x]^m, x]; Table[T[[n-k+1, k]], {n, 1, m}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Mar 20 2019 *)

A362563 Triangle T(n, k) read by rows, where T(n, k) is the number of {123,132}-avoiding parking functions of size n with k active sites, for 2 <= k <= n+1.

Original entry on oeis.org

1, 1, 2, 1, 3, 4, 3, 5, 8, 8, 8, 14, 17, 20, 16, 24, 40, 49, 50, 48, 32, 75, 123, 147, 151, 136, 112, 64, 243, 393, 465, 473, 432, 352, 256, 128, 808, 1294, 1519, 1540, 1409, 1176, 880, 576, 256, 2742, 4358, 5087, 5144, 4721, 3986, 3088, 2144, 1280, 512

Offset: 1

Lara Pudwell, Apr 24 2023

Consider a parking function of size n that avoids both 123 and 132.
Such a parking function can be represented as a labeled Dyck path (using steps N = (0, 1) and E = (1, 0) staying weakly above y = x), where the north steps are labeled with 1, 2, ..., n, and where consecutive north steps have increasing labels.
An active site is a point where the parking function's corresponding Dyck path touches y = x.
T(n, k) is the number of parking functions of size n with exactly k active sites.

			Triangle T(n, k) begins:
     1;
     1,    2;
     1,    3,    4;
     3,    5,    8,    8;
     8,   14,   17,   20,   16;
    24,   40,   49,   50,   48,   32;
    75,  123,  147,  151,  136,  112,   64;
   243,  393,  465,  473,  432,  352,  256,  128;
   808, 1294, 1519, 1540, 1409, 1176,  880,  576,  256;
  2742, 4358, 5087, 5144, 4721, 3986, 3088, 2144, 1280, 512;
  ...
The eight {123,132}-avoiding parking functions of size 3 are 211, 212, 213, 221, 231, 311, 312, and 321.
In block notation:
   211 is {2,3},{1},{}  -> NNENEE, which has 2 active sites;
   212 is {2},{1, 3},{} -> NENNEE, which has 3 active sites;
   213 is {2},{1},{3}   -> NENENE, which has 4 active sites;
   221 is {3},{1,2},{}  -> NENNEE, which has 3 active sites;
   231 is {3},{1},{2}   -> NENENE, which has 4 active sites;
   311 is {2,3},{},{1}  -> NNEENE, which has 3 active sites;
   312 is {2},{3},{1}   -> NENENE, which has 4 active sites;
   321 is {3},{2},{1}   -> NENENE, which has 4 active sites.
So T(3,2) = 1, T(3,3) = 3, T(3,4) = 4.

Ayomikun Adeniran and Lara Pudwell, Pattern avoidance in parking functions, Enumer. Comb. Appl. 3:3 (2023), Article S2R17.

Cf. A000079 (right diagonal), A000958 (1st column and row sums).

T(n, k) = 0 if k < 2 or k > n+1
T(1, 2) = T(2, 2) = 1.
T(2, 3) = 2.
For n > 2, T(n, k) = 2*T(n-1, k-1) + Sum_{j=k-1..n-1} T(n-2, j).
T(n, n+1) = A000079(n-1).
Sum_{k=2..n+1} T(n, k) = T(n+2, 2) = A000958(n+1).

cp's OEIS Frontend

A127539 Number of ordered trees with n edges having no odd-length branches starting at the root.

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A127541 Triangle read by rows: T(n,k) is the number of ordered trees with n edges having k even-length branches starting at the root (0<=k<=n).

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A167769 Pendular trinomial triangle (p=0), read by rows of 2n+1 terms (n>=0), defined by the recurrence : if 0

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A193215 Number of Dyck paths of semilength n having the property that the heights of the first and the last peaks coincide.

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A236918 Triangle read by rows: Catalan triangle of the k-Fibonacci sequence.

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A237619 Riordan array (1/(1+x*c(x)), x*c(x)) where c(x) is the g.f. of Catalan numbers (A000108).

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A098747 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n having exactly k UDU's at low level.

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A237619 Riordan array (1/(1+xc(x)), xc(x)) where c(x) is the g.f. of Catalan numbers (A000108).

A323212 The Fibonacci-Catalan Hybrid. Expansion of 1 + x(2(x + 1))/(sqrt(1 - 4y) - 2x*(x + 1) + 1). Square array read by descending antidiagonals, A(n,k) for n,k >= 0.