A026781 -id:A026781 - cp's OEIS frontend

A026780 Triangular array T read by rows: T(n,0)=T(n,n)=1 for n >= 0; for n >= 2 and 1 <= k <= n-1, T(n,k) = T(n-1,k-1) + T(n-2,k-1) + T(n-1,k) if 1 <= k <= floor(n/2), else T(n,k) = T(n-1,k-1) + T(n-1,k).

1, 1, 1, 1, 3, 1, 1, 5, 4, 1, 1, 7, 12, 5, 1, 1, 9, 24, 17, 6, 1, 1, 11, 40, 53, 23, 7, 1, 1, 13, 60, 117, 76, 30, 8, 1, 1, 15, 84, 217, 246, 106, 38, 9, 1, 1, 17, 112, 361, 580, 352, 144, 47, 10, 1, 1, 19, 144, 557, 1158, 1178, 496, 191, 57, 11, 1

Offset: 0

Clark Kimberling

T(n,k) is the number of paths from (0,0) to (k,n-k) in the directed graph having vertices (i,j) and edges (i,j)-to-(i+1,j) and (i,j)-to-(i,j+1) for i,j>= 0 and edges (i,i+h)-to-(i+1,i+h+1) for i>=0, h>=0.

Also, square array R read by antidiagonals with R(i,j) = T(i+j,i) equal number of paths from (0,0) to (i,j). - Max Alekseyev, Jan 13 2015

			The array T(n,k) starts with:
n=0: 1;
n=1: 1,  1;
n=2: 1,  3,  1;
n=3: 1,  5,  4,  1;
n=4: 1,  7, 12,  5,  1;
n=5: 1,  9, 24, 17,  6, 1;
n=6: 1, 11, 40, 53, 23, 7, 1;
...

G. C. Greubel, Rows n = 0..100 of triangle, flattened
M. A. Alekseyev, On Enumeration of Dyck-Schroeder Paths, Journal of Combinatorial Mathematics and Combinatorial Computing 106 (2018), 59-68; arXiv:1601.06158 [math.CO], 2016-2018.

Cf. A026787 (row sums), A026781 (center elements), A249488 (row-reversed version).

Cf. A026782, A026783, A026784, A026785, A026786, A026787, A026788, A026789, A026790.

T:= function(n,k)
    if n<0 then return 0;
    elif k=0 or k=n then return 1;
    elif (k <= Int(n/2)) then return T(n-1,k-1)+T(n-2,k-1) +T(n-1,k);
    else return T(n-1,k-1) + T(n-1,k);
    fi;
  end;
Flat(List([0..12], n-> List([0..n], k-> T(n,k) ))); # G. C. Greubel, Oct 31 2019

T:= proc(n,k) option remember;
    if n<0 then 0;
    elif k=0 or k =n then 1;
    elif k <= n/2 then
        procname(n-1,k-1)+procname(n-2,k-1)+procname(n-1,k) ;
    else
        procname(n-1,k-1)+procname(n-1,k) ;
    fi ;
end proc:
seq(seq(T(n,k), k=0..n), n=0..12); # G. C. Greubel, Nov 01 2019

T[n_, k_]:= T[n, k]= If[n<0, 0, If[k==0 || k==n, 1, If[k<=n/2, T[n-1, k-1] + T[n-2, k-1] + T[n-1, k], T[n-1, k-1] + T[n-1, k] ]]];
Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Nov 01 2019 *)

T(n,k) = if(n<0, 0, if(k==0 || k==n, 1, if( k<=n/2, T(n-1,k-1) + T(n-2,k-1) + T(n-1,k), T(n-1,k-1) + T(n-1,k) ));)
for(n=0,12, for(k=0,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Oct 31 2019

@CachedFunction
def T(n, k):
    if (n<0): return 0
    elif (k==0 or k==n): return 1
    elif (k<=n/2): return T(n-1,k-1) + T(n-2,k-1) + T(n-1,k)
    else: return T(n-1,k-1) + T(n-1,k)
[[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Oct 31 2019

For n>=2*k, T(n,k) = coefficient of x^k in F(x)*S(x)^(n-2*k). For n<=2*k, T(n,k) = coefficient of x^(n-k) in F(x)*C(x)^(2*k-n). Here C(x) = (1 - sqrt(1-4x))/(2*x) is o.g.f. for A000108, S(x) = (1 - x - sqrt(1-6*x+x^2))/(2*x) is o.g.f. for A006318, and F(x) = S(x)/(1 - x*C(x)*S(x)) is o.g.f. for A026781. - Max Alekseyev, Jan 13 2015

Edited by Max Alekseyev, Dec 02 2015

A026671 Number of lattice paths from (0,0) to (n,n) with steps (0,1), (1,0) and, when on the diagonal, (1,1).

Original entry on oeis.org

1, 3, 11, 43, 173, 707, 2917, 12111, 50503, 211263, 885831, 3720995, 15652239, 65913927, 277822147, 1171853635, 4945846997, 20884526283, 88224662549, 372827899079, 1576001732485, 6663706588179, 28181895551161, 119208323665543, 504329070986033, 2133944799315027

Offset: 0

Clark Kimberling, Miklos Bona

1, 1, 3, 11, 43, 173, ... is the unique sequence for which both the Hankel transform of the sequence itself and the Hankel transform of its left shift are the powers of 2 (A000079). For example, det[{{1, 1, 3}, {1, 3, 11}, {3, 11, 43}}] = det[{{1, 3, 11}, {3, 11, 43}, {11, 43, 173}}] = 4. - David Callan, Mar 30 2007
From Paul Barry, Jan 25 2009: (Start)
a(n) is the image of F(2n+2) under the Catalan matrix (1,xc(x)) where c(x) is the g.f. of A000108.
The sequence 1,1,3,... is the image of A001519 under (1,xc(x)). This sequence has g.f. given by 1/(1-x-2x^2/(1-3x-x^2/(1-2x-x^2/(1-2x-x^2/(1-... (continued fraction). (End)
Binomial transform of A111961. - Philippe Deléham, Feb 11 2009
From Paul Barry, Nov 03 2010: (Start)
The sequence 1,1,3,... has g.f. 1/(1-x/sqrt(1-4x)), INVERT transform of A000984.
It is an eigensequence of the sequence array for A000984. (End)

L. W. Shapiro and C. J. Wang, Generating identities via 2 X 2 matrices, Congressus Numerantium, 205 (2010), 33-46.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Jean-Christophe Aval, Adrien Boussicault and Sandrine Dasse-Hartaut, The tree structure in staircase tableaux, arXiv:1109.4907 [math.CO], 2011-2013.
Cyril Banderier, Markus Kuba, and Michael Wallner, Analytic Combinatorics of Composition schemes and phase transitions with mixed Poisson distributions, arXiv:2103.03751 [math.PR], 2021.
Miklos Bona, The permutation classes equinumerous to the smooth class, Electron. J. Combin., 5 (1998), no. 1, Research Paper 31, 12 pp.
David Callan and Toufik Mansour, Five subsets of permutations enumerated as weak sorting permutations, arXiv:1602.05182 [math.CO], 2016.
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.
Milan Janjić, Pascal Matrices and Restricted Words, J. Int. Seq., Vol. 21 (2018), Article 18.5.2.
J. W. Layman, The Hankel Transform and Some of its Properties, J. Integer Sequences, 4 (2001), #01.1.5.
Huyile Liang, Jeffrey Remmel and Sainan Zheng, Stieltjes moment sequences of polynomials, arXiv:1710.05795 [math.CO], 2017, see page 16.

a(n) = T(2n-1, n-1), T given by A026736.
a(n) = T(2n, n), T given by A026670.
a(n) = T(2n+1, n+1), T given by A026725.
Row sums of triangle A054335.
Cf. A026781, A001076, A176280.

a:=[3,11,43];; for n in [4..30] do a[n]:=(2*(4*n-3)*a[n-1] - 3*(5*n-8)*a[n-2] - 2*(2*n-3)*a[n-3])/n; od; Concatenation([1], a); # G. C. Greubel, Jul 16 2019

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( 1/(Sqrt(1-4*x)-x) )); // G. C. Greubel, Jul 16 2019

Table[SeriesCoefficient[1/(Sqrt[1-4*x]-x),{x,0,n}],{n,0,30}] (* Vaclav Kotesovec, Oct 08 2012 *)

{a(n)= if(n<0, 0, polcoeff( 1/(sqrt(1 -4*x +x*O(x^n)) -x), n))} /* Michael Somos, Apr 20 2007 */

my(x='x+O('x^66)); Vec( 1/(sqrt(1-4*x)-x) ) \\ Joerg Arndt, May 04 2013

(1/(sqrt(1-4*x)-x)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jul 16 2019

From Wolfdieter Lang, Mar 21 2000: (Start)
G.f.: 1/(sqrt(1-4*x)-x).
a(n) = Sum_{i=1..n} a(i-1)*binomial(2*(n-i), n-i) + binomial(2*n, n), n >= 1, a(0)=1. (End)
G.f.: 1/(1 -x -2*x*c(x)) where c(x) = g.f. for Catalan numbers A000108. - Michael Somos, Apr 20 2007
From Paul Barry, Jan 25 2009: (Start)
G.f.: 1/(1 - 3xc(x) + x^2*c(x)^2);
G.f.: 1/(1-3x-2x^2/(1-2x-x^2/(1-2x-x^2/(1-2x-x^2/(1-... (continued fraction).
a(0) = 1, a(n) = Sum_{k=0..n} (k/(2n-k))*C(2n-k,n-k)*F(2k+2). (End)
a(n) = Sum_{k=0..n} A039599(n,k) * A000045(k+2). - Philippe Deléham, Feb 11 2009
From Paul Barry, Feb 08 2009: (Start)
G.f.: 1/(1-x/(1-2x/(1-x/(1-x/(1-x/(1-x/(1-x/(1-... (continued fraction);
G.f. of 1,1,3,... is 1/(1-x-2x/(1-x/(1-x/(1-x/(1-... (continued fraction). (End)
From Gary W. Adamson, Jul 14 2011: (Start)
a(n) = the upper left term in M^n, M = the infinite square production matrix:
  3, 2, 0, 0, 0, 0, ...
  1, 1, 1, 0, 0, 0, ...
  1, 1, 1, 1, 0, 0, ...
  1, 1, 1, 1, 1, 0, ...
  1, 1, 1, 1, 1, 1, ...
  ... (End)
From Vaclav Kotesovec, Oct 08 2012: (Start)
D-finite with recurrence: n*a(n) = 2*(4*n-3)*a(n-1) - 3*(5*n-8)*a(n-2) - 2*(2*n-3)*a(n-3).
a(n) ~ (2+sqrt(5))^n/sqrt(5). (End)
a(n) = Sum_{k=0..n+1} 4^(n+1-k) * binomial(n-k/2,n+1-k). - Seiichi Manyama, Mar 30 2025
From Peter Luschny, Mar 30 2025: (Start)
a(n) = 4^n*(binomial(n-1/2, n)*hypergeom([1, (1-n)/2, -n/2], [1/2, 1/2-n], -1/4) + hypergeom([(1-n)/2, 1-n/2], [1-n], -1/4)/4) for n > 0.
a(n) = A001076(n) + A176280(n). (End)

A026787 a(n) = Sum_{k=0..n} T(n,k), T given by A026780.

Original entry on oeis.org

1, 2, 5, 11, 26, 58, 136, 306, 717, 1625, 3813, 8697, 20451, 46909, 110563, 254855, 602042, 1393746, 3299304, 7666786, 18182976, 42391546, 100704606, 235452416, 560147414, 1312916040, 3127406812, 7346213746, 17518138314, 41228281888, 98408997716, 231990850378, 554207752781, 1308436686305, 3128033585157

Offset: 0

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A026780, A026781, A026782, A026783, A026784, A026785, A026786, A026788, A026789, A026790.

T:= proc(n,k) option remember;
    if n<0 then 0;
    elif k=0 or k =n then 1;
    elif k <= n/2 then
        procname(n-1,k-1)+procname(n-2,k-1)+procname(n-1,k) ;
    else
        procname(n-1,k-1)+procname(n-1,k) ;
    fi ;
end proc:
seq( add(T(n,k), k=0..n), n=0..30); # G. C. Greubel, Nov 02 2019

T[n_, k_]:= T[n, k]= If[n<0, 0, If[k==0 || k==n, 1, If[k<=n/2, T[n-1, k-1] + T[n-2, k-1] + T[n-1, k], T[n-1, k-1] + T[n-1, k] ]]];
Table[Sum[T[n, k], {k,0,n}], {n,0,30}] (* G. C. Greubel, Nov 02 2019 *)

@CachedFunction
def T(n, k):
    if (n<0): return 0
    elif (k==0 or k==n): return 1
    elif (k<=n/2): return T(n-1,k-1) + T(n-2,k-1) + T(n-1,k)
    else: return T(n-1,k-1) + T(n-1,k)
[sum(T(n,k) for k in (0..n)) for n in (0..30)] # G. C. Greubel, Nov 02 2019

O.g.f.: F(x^2)*(1/(1-x*S(x^2))+C(x^2)*x/(1-x*C(x^2))), where C(x)=(1-sqrt(1-4x))/(2*x) is o.g.f. for A000108, S(x)=(1-x-sqrt(1-6*x+x^2))/(2*x) is o.g.f. for A006318, and F(x)=S(x)/(1-x*C(x)*S(x)) is o.g.f. for A026781. - Max Alekseyev, Jan 13 2015

C(x^2)/(1-x*C(x^2)) above is the o.g.f. for A001405. 1/(1-x*S(x^2)) above is the o.g.f for A026003 starting with an additional 1: 1,1,1,3,5,13,25,... - R. J. Mathar, Feb 10 2022

More terms from Max Alekseyev, Jan 13 2015

A026770 a(n) = T(2n,n), T given by A026769.

Original entry on oeis.org

1, 2, 7, 28, 120, 538, 2493, 11854, 57558, 284392, 1426038, 7241356, 37173304, 192638992, 1006564439, 5297715628, 28061959428, 149491856978, 800425486692, 4305263668514, 23251846197766, 126044501870378, 685569373724964, 3740339567665558, 20463965229643218, 112250484320225118

Offset: 0

Clark Kimberling

nonn

Number of lattice paths from (0,0) to (n,n) with steps (0,1), (1,0) and, when below the diagonal, (1,1). - Alois P. Heinz, Sep 14 2016

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A000108, A006318, A026781, A104625, A109980.

Cf. A026769, A026771, A026772, A026773, A026774, A026775, A026776, A026777, A026778, A026779.

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( 2/(x + Sqrt(1-4*x) + Sqrt(1-6*x+x^2)) )); // G. C. Greubel, Nov 01 2019

seq(coeff(series(2/(x + sqrt(1-4*x) + sqrt(1-6*x+x^2)), x, n+1), x, n), n = 0..30); # G. C. Greubel, Nov 01 2019

T[n_, k_] := T[n, k] = Which[k==0 || k==n, 1, n==2 && k==1, 2, k<=(n-1)/2, T[n-1, k-1] + T[n-2, k-1] + T[n-1, k], True, T[n-1, k-1] + T[n-1, k]];
a[n_] := T[2n, n];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, May 24 2019 *)

{ C = (1-sqrt(1-4*x+O(x^51)))/2/x; S = (1-x-sqrt(1-6*x+x^2 +O(x^51)))/2/x; Vec(1/(1-x*(C+S))) } /* Max Alekseyev, Dec 02 2015 */

def A026770_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 2/(x + sqrt(1-4*x) + sqrt(1-6*x+x^2)) ).list()
A026770_list(30) # G. C. Greubel, Nov 01 2019

O.g.f.: 1/(1-x*(C(x)+S(x))), where C(x)=(1-sqrt(1-4x))/(2*x) is o.g.f. for A000108 and S(x)=(1-x-sqrt(1-6*x+x^2))/(2*x) is o.g.f. for A006318. - Max Alekseyev, Dec 02 2015

A026782 a(n) = T(2n,n-1), T given by A026780.

Original entry on oeis.org

1, 7, 40, 217, 1158, 6150, 32656, 173719, 926664, 4958556, 26619438, 143365880, 774562478, 4197344582, 22810572062, 124300860689, 679081142350, 3718894341450, 20412141531664, 112276061739814, 618806031336236, 3416954495002676

Offset: 1

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 1..500

Cf. A026780, A026781, A026783, A026784, A026785, A026786, A026787, A026788, A026789, A026790.

T:= proc(n,k) option remember;
    if n<0 then 0;
    elif k=0 or k =n then 1;
    elif k <= n/2 then
        procname(n-1,k-1)+procname(n-2,k-1)+procname(n-1,k) ;
    else
        procname(n-1,k-1)+procname(n-1,k) ;
    fi ;
end proc:
seq(T(2*n,n-1), n=1..30); # G. C. Greubel, Nov 02 2019

T[n_, k_]:= T[n, k]= If[n<0, 0, If[k==0 || k==n, 1, If[k<=n/2, T[n-1, k-1] + T[n-2, k-1] + T[n-1, k], T[n-1, k-1] + T[n-1, k] ]]];
Table[T[2*n, n-1], {n, 30}] (* G. C. Greubel, Nov 02 2019 *)

@CachedFunction
def T(n, k):
    if (n<0): return 0
    elif (k==0 or k==n): return 1
    elif (k<=n/2): return T(n-1,k-1) + T(n-2,k-1) + T(n-1,k)
    else: return T(n-1,k-1) + T(n-1,k)
[T(2*n, n-1) for n in (1..30)] # G. C. Greubel, Nov 02 2019

A026783 a(n) = T(2n, n-2), T given by A026780.

Original entry on oeis.org

1, 11, 84, 557, 3446, 20514, 119336, 684227, 3886460, 21939528, 123347842, 691644044, 3871738018, 21652138770, 121026492186, 676391629701, 3780636102222, 21137831159462, 118234019051048, 661686074145618, 3705252204960252

Offset: 2

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 2..500

Cf. A026780, A026781, A026782, A026784, A026785, A026786, A026787, A026788, A026789, A026790.

T:= proc(n,k) option remember;
    if n<0 then 0;
    elif k=0 or k =n then 1;
    elif k <= n/2 then
        procname(n-1,k-1)+procname(n-2,k-1)+procname(n-1,k) ;
    else
        procname(n-1,k-1)+procname(n-1,k) ;
    fi ;
end proc:
seq(T(2*n,n-2), n=2..30); # G. C. Greubel, Nov 02 2019

T[n_, k_]:= T[n, k]= If[n<0, 0, If[k==0 || k==n, 1, If[k<=n/2, T[n-1, k-1] + T[n-2, k-1] + T[n-1, k], T[n-1, k-1] + T[n-1, k] ]]];
Table[T[2*n, n-2], {n,2,30}] (* G. C. Greubel, Nov 02 2019 *)

@CachedFunction
def T(n, k):
    if (n<0): return 0
    elif (k==0 or k==n): return 1
    elif (k<=n/2): return T(n-1,k-1) + T(n-2,k-1) + T(n-1,k)
    else: return T(n-1,k-1) + T(n-1,k)
[T(2*n, n-2) for n in (2..30)] # G. C. Greubel, Nov 02 2019

A026784 a(n) = T(2n-1, n-1), T given by A026780.

Original entry on oeis.org

1, 5, 24, 117, 580, 2916, 14834, 76221, 395048, 2063104, 10847078, 57373672, 305110106, 1630489090, 8751851866, 47166202181, 255128842340, 1384688987728, 7538592535170, 41159292861980, 225315261459390, 1236441650047554

Offset: 1

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 1..500

Cf. A026780, A026781, A026782, A026783, A026785, A026786, A026787, A026788, A026789, A026790.

T:= proc(n,k) option remember;
    if n<0 then 0;
    elif k=0 or k =n then 1;
    elif k <= n/2 then
        procname(n-1,k-1)+procname(n-2,k-1)+procname(n-1,k) ;
    else
        procname(n-1,k-1)+procname(n-1,k) ;
    fi ;
end proc:
seq(T(2*n-1,n-1), n=1..30); # G. C. Greubel, Nov 02 2019

T[n_, k_]:= T[n, k]= If[n<0, 0, If[k==0 || k==n, 1, If[k<=n/2, T[n-1, k-1] + T[n-2, k-1] + T[n-1, k], T[n-1, k-1] + T[n-1, k] ]]];
Table[T[2*n-1, n-1], {n, 30}] (* G. C. Greubel, Nov 02 2019 *)

@CachedFunction
def T(n, k):
    if (n<0): return 0
    elif (k==0 or k==n): return 1
    elif (k<=n/2): return T(n-1,k-1) + T(n-2,k-1) + T(n-1,k)
    else: return T(n-1,k-1) + T(n-1,k)
[T(2*n-1, n-1) for n in (1..30)] # G. C. Greubel, Nov 02 2019

A026785 a(n) = T(2n-1, n-2), T given by A026780.

Original entry on oeis.org

1, 9, 60, 361, 2076, 11672, 64842, 357897, 1968788, 10813804, 59372770, 326086492, 1792293014, 9861375614, 54324086446, 299651439321, 1655124211372, 9154654655044, 50704627346170, 281214708137032, 1561706813618886

Offset: 2

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 2..500

Cf. A026780, A026781, A026782, A026783, A026784, A026786, A026787, A026788, A026789, A026790.

T:= proc(n,k) option remember;
    if n<0 then 0;
    elif k=0 or k =n then 1;
    elif k <= n/2 then
        procname(n-1,k-1)+procname(n-2,k-1)+procname(n-1,k) ;
    else
        procname(n-1,k-1)+procname(n-1,k) ;
    fi ;
end proc:
seq(T(2*n-1,n-2), n=2..30); # G. C. Greubel, Nov 02 2019

T[n_, k_]:= T[n, k]= If[n<0, 0, If[k==0 || k==n, 1, If[k<=n/2, T[n-1, k-1] + T[n-2, k-1] + T[n-1, k], T[n-1, k-1] + T[n-1, k] ]]];
Table[T[2*n-1, n-2], {n,2,30}] (* G. C. Greubel, Nov 02 2019 *)

@CachedFunction
def T(n, k):
    if (n<0): return 0
    elif (k==0 or k==n): return 1
    elif (k<=n/2): return T(n-1,k-1) + T(n-2,k-1) + T(n-1,k)
    else: return T(n-1,k-1) + T(n-1,k)
[T(2*n-1, n-2) for n in (2..30)] # G. C. Greubel, Nov 02 2019

A026786 a(n) = T(n, floor(n/2)), T given by A026780.

Original entry on oeis.org

1, 1, 3, 5, 12, 24, 53, 117, 246, 580, 1178, 2916, 5768, 14834, 28731, 76221, 145108, 395048, 741392, 2063104, 3825418, 10847078, 19907156, 57373672, 104370554, 305110106, 550816506, 1630489090, 2924018194, 8751851866

Offset: 0

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A026780, A026781, A026782, A026783, A026784, A026785, A026787, A026788, A026789, A026790.

T:= proc(n,k) option remember;
    if n<0 then 0;
    elif k=0 or k =n then 1;
    elif k <= n/2 then
        procname(n-1,k-1)+procname(n-2,k-1)+procname(n-1,k) ;
    else
        procname(n-1,k-1)+procname(n-1,k) ;
    fi ;
end proc:
seq(T(n, floor(n/2)), n=0..30); # G. C. Greubel, Nov 02 2019

T[n_, k_]:= T[n, k]= If[n<0, 0, If[k==0 || k==n, 1, If[k<=n/2, T[n-1, k-1] + T[n-2, k-1] + T[n-1, k], T[n-1, k-1] + T[n-1, k] ]]];
Table[T[n, Floor[n/2]], {n,0,30}] (* G. C. Greubel, Nov 02 2019 *)

@CachedFunction
def T(n, k):
    if (n<0): return 0
    elif (k==0 or k==n): return 1
    elif (k<=n/2): return T(n-1,k-1) + T(n-2,k-1) + T(n-1,k)
    else: return T(n-1,k-1) + T(n-1,k)
[T(n, floor(n/2)) for n in (0..30)] # G. C. Greubel, Nov 02 2019

A026788 a(n) = Sum_{k=0..floor(n/2)} T(n,k), T given by A026780.

Original entry on oeis.org

1, 1, 4, 6, 20, 34, 105, 191, 563, 1071, 3057, 6007, 16745, 33729, 92332, 189662, 511812, 1068178, 2849404, 6025594, 15921514, 34043204, 89242582, 192621212, 501574732, 1091400122, 2825710822, 6192005260, 15952433940, 35172854946

Offset: 0

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A026780, A026781, A026782, A026783, A026784, A026785, A026786, A026787, A026789, A026790.

T:= proc(n,k) option remember;
    if n<0 then 0;
    elif k=0 or k =n then 1;
    elif k <= n/2 then
        procname(n-1,k-1)+procname(n-2,k-1)+procname(n-1,k) ;
    else
        procname(n-1,k-1)+procname(n-1,k) ;
    fi ;
end proc:
seq( add(T(n,k), k=0..floor(n/2)), n=0..30); # G. C. Greubel, Nov 02 2019

T[n_, k_]:= T[n, k]= If[n<0, 0, If[k==0 || k==n, 1, If[k<=n/2, T[n-1, k-1] + T[n-2, k-1] + T[n-1, k], T[n-1, k-1] + T[n-1, k] ]]];
Table[Sum[T[n, k], {k, 0, Floor[n/2]}], {n,0,30}] (* G. C. Greubel, Nov 02 2019 *)

@CachedFunction
def T(n, k):
    if (n<0): return 0
    elif (k==0 or k==n): return 1
    elif (k<=n/2): return T(n-1,k-1) + T(n-2,k-1) + T(n-1,k)
    else: return T(n-1,k-1) + T(n-1,k)
[sum(T(n,k) for k in (0..floor(n/2))) for n in (0..30)] # G. C. Greubel, Nov 02 2019

cp's OEIS Frontend

A026780 Triangular array T read by rows: T(n,0)=T(n,n)=1 for n >= 0; for n >= 2 and 1 <= k <= n-1, T(n,k) = T(n-1,k-1) + T(n-2,k-1) + T(n-1,k) if 1 <= k <= floor(n/2), else T(n,k) = T(n-1,k-1) + T(n-1,k).

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A026671 Number of lattice paths from (0,0) to (n,n) with steps (0,1), (1,0) and, when on the diagonal, (1,1).

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A026787 a(n) = Sum_{k=0..n} T(n,k), T given by A026780.

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A026770 a(n) = T(2n,n), T given by A026769.

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A026782 a(n) = T(2n,n-1), T given by A026780.

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A026783 a(n) = T(2n, n-2), T given by A026780.

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A026784 a(n) = T(2n-1, n-1), T given by A026780.