A026736 -id:A026736 - cp's OEIS frontend

A027217 a(n) = Sum_{k=0..n-2} T(n,k)*T(n,k+2), T given by A026736.

1, 6, 32, 136, 640, 2593, 11860, 47532, 215531, 861334, 3893621, 15549166, 70199065, 280316029, 1264697307, 5050617474, 22776900816, 90972831448, 410117333080

Offset: 2

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 2..1000

Cf. A026736.

T:= function(n, k)
    if k=0 or k=n then return 1;
    elif k=n-1 then return n;
    elif (n mod 2)=0 and k=Int((n-2)/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;
List([2..20], n-> Sum([0..n-2], k-> T(n, k)*T(n,k+2) )); # G. C. Greubel, Jul 19 2019

T[n_, k_]:= T[n, k] = If[k==0 || k==n, 1, If[EvenQ[n] && k==(n-2)/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]*T[n,k+2], {k, 0, n-2}], {n, 2, 30}] (* G. C. Greubel, Jul 19 2019 *)

T(n, k) = if(k==n || k==0, 1, k==n-1, n, if((n%2)==0 && k==(n-2)/2, T(n-1, k-1) + T(n-2, k-1) + T(n-1, k), T(n-1, k-1) + T(n-1, k) ));
vector(20, n, n++; sum(k=0, n-2, T(n, k)*T(n,k+2)) ) \\ G. C. Greubel, Jul 19 2019

@CachedFunction
def T(n, k):
    if (k==0 or k==n): return 1
    elif (mod(n, 2)==0 and k==(n-2)/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)*T(n,k+2) for k in (0..n-2)) for n in (2..30)] # G. C. Greubel, Jul 19 2019

A027218 a(n) = Sum_{k=0..n-3} T(n,k)*T(n,k+3), T given by A026736.

Original entry on oeis.org

1, 9, 51, 279, 1277, 6235, 26789, 125370, 525082, 2409886, 9969722, 45289767, 186105280, 840402559, 3439358196, 15472942142, 63155131233, 283400162019

Offset: 3

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 3..1000

Cf. A026736.

T:= function(n, k)
    if k=0 or k=n then return 1;
    elif k=n-1 then return n;
    elif (n mod 2)=0 and k=Int((n-2)/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;
List([3..20], n-> Sum([0..n-3], k-> T(n, k)*T(n,k+3) )); # G. C. Greubel, Jul 19 2019

T[n_, k_]:= T[n, k] = If[k==0 || k==n, 1, If[EvenQ[n] && k==(n-2)/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]*T[n,k+3], {k, 0, n-3}], {n, 3, 30}] (* G. C. Greubel, Jul 19 2019 *)

T(n, k) = if(k==n || k==0, 1, k==n-1, n, if((n%2)==0 && k==(n-2)/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=3,20, print1(sum(k=0, n-3, T(n, k)*T(n,k+3)), ", ")) \\ G. C. Greubel, Jul 19 2019

@CachedFunction
def T(n, k):
    if (k==0 or k==n): return 1
    elif (mod(n, 2)==0 and k==(n-2)/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)*T(n,k+3) for k in (0..n-3)) for n in (3..30)] # G. C. Greubel, Jul 19 2019

A027219 a(n) = Sum_{k=0..n} (k+1) * A026736(n,k).

Original entry on oeis.org

1, 3, 8, 20, 50, 117, 283, 639, 1512, 3338, 7774, 16898, 38884, 83566, 190488, 405848, 918120, 1942813, 4367665, 9191499, 20555546, 43061789, 95874233, 200083005, 443770612, 923124007, 2040635445, 4233080627, 9330343290

Offset: 0

Clark Kimberling

nonn

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

Cf. A026736.

T:= function(n, k)
    if k=0 or k=n then return 1;
    elif k=n-1 then return n;
    elif (n mod 2)=0 and k=Int((n-2)/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;
List([0..20], n-> Sum([0..n], k-> (k+1)*T(n, k) )); # G. C. Greubel, Jul 19 2019

T[n_, k_]:= T[n, k] = If[k==0 || k==n, 1, If[EvenQ[n] && k==(n-2)/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[(k+1)*T[n,k], {k, 0, n}], {n, 0, 30}] (* G. C. Greubel, Jul 19 2019 *)

T(n, k) = if(k==n || k==0, 1, k==n-1, n, if((n%2)==0 && k==(n-2)/2, T(n-1, k-1) + T(n-2, k-1) + T(n-1, k), T(n-1, k-1) + T(n-1, k) ));
vector(20, n, n--; sum(k=0, n, (k+1)*T(n, k)) ) \\ G. C. Greubel, Jul 19 2019

@CachedFunction
def T(n, k):
    if (k==0 or k==n): return 1
    elif (mod(n, 2)==0 and k==(n-2)/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((k+1)*T(n,k) for k in (0..n)) for n in (0..30)] # G. C. Greubel, Jul 19 2019

A027220 a(n) = Sum_{k=0..n} (k+1) * A026736(n,n-k).

Original entry on oeis.org

1, 3, 8, 20, 52, 121, 301, 675, 1628, 3570, 8426, 18202, 42288, 90374, 207464, 439800, 1000194, 2106961, 4755715, 9967599, 22359788, 46670273, 104154703, 216643945, 481381746, 998346275, 2210037191, 4571884119, 10088030640

Offset: 0

Clark Kimberling

nonn

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

Cf. A026736.

T[n_, k_]:= T[n, k] = If[k==0 || k==n, 1, If[EvenQ[n] && k==(n-2)/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[(k+1)*T[n,n-k], {k, 0, n}], {n, 0, 30}] (* G. C. Greubel, Jul 19 2019 *)

@CachedFunction
def T(n, k):
    if (k==0 or k==n): return 1
    elif (mod(n, 2)==0 and k==(n-2)/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((k+1)*T(n,n-k) for k in (0..n)) for n in (0..30)] # G. C. Greubel, Jul 19 2019

A027221 Sum of squares of numbers in row n of array T given by A026736.

Original entry on oeis.org

1, 2, 6, 20, 79, 284, 1237, 4542, 20626, 76406, 354080, 1317964, 6173634, 23051344, 108628550, 406513364, 1922354351, 7206349304, 34147706833, 128187589014, 608151037123, 2285559568866, 10850577045131, 40817923301712, 193850277807569, 729825857819924, 3466587141136257

Offset: 0

Clark Kimberling

nonn

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

Cf. A026736.

T:= function(n, k)
    if k=0 or k=n then return 1;
    elif k=n-1 then return n;
    elif (n mod 2)=0 and k=Int((n-2)/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;
List([0..21], n-> Sum([0..n], k-> T(n, k)^2 )); # G. C. Greubel, Jul 19 2019

T[n_, k_]:= T[n, k] = If[k==0 || k==n, 1, If[EvenQ[n] && k==(n-2)/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]^2, {k,0,n}], {n,0,40}] (* G. C. Greubel, Jul 19 2019 *)

T(n, k) = if(k==n || k==0, 1, k==n-1, n, if((n%2)==0 && k==(n-2)/2, T(n-1, k-1) + T(n-2, k-1) + T(n-1, k), T(n-1, k-1) + T(n-1, k) ));
vector(21, n, n--; sum(k=0, n, T(n, k)^2 ) ) \\ G. C. Greubel, Jul 19 2019

@CachedFunction
def T(n, k):
    if (k==0 or k==n): return 1
    elif (mod(n, 2)==0 and k==(n-2)/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)^2 for k in (0..n)) for n in (0..40)] # G. C. Greubel, Jul 19 2019

A027214 a(n) = greatest number in row n of array T given by A026736.

Original entry on oeis.org

1, 1, 2, 3, 6, 11, 22, 43, 94, 173, 398, 707, 1680, 2917, 7085, 12111, 29877, 50503, 126021, 211263, 531751, 885831, 2244627, 3720995, 9478605, 15652239, 40040183, 65913927, 169193597, 277822147, 715143046, 1171853635, 3023492646

Offset: 0

Clark Kimberling

nonn

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)

A111279 Number of permutations avoiding the patterns {3241,3421,4321}; number of weak sorting class based on 3241.

Original entry on oeis.org

1, 1, 2, 6, 21, 79, 309, 1237, 5026, 20626, 85242, 354080, 1476368, 6173634, 25873744, 108628550, 456710589, 1922354351, 8098984433, 34147706833, 144068881455, 608151037123, 2568318694867, 10850577045131, 45856273670841, 193850277807569, 819669810565949

Offset: 0

Len Smiley, Nov 01 2005

nonn

Is this the same sequence as A026737? - Andrew S. Plewe, May 09 2007
Yes, see the Callan reference "A bijection...". - Joerg Arndt, Feb 29 2016
a(n) is the number of permutations of length n avoiding the partially ordered pattern (POP) {1>3, 1>4, 3>2} of length 4. That is, the number of length n permutations having no subsequences of length 4 in which the first element is the largest and the third element is larger than the second element. - Sergey Kitaev, Dec 10 2020

			a(4) = 21 since the top row terms of M^3 = (11, 6, 3, 1, 0, 0, 0, ...)

Alois P. Heinz, Table of n, a(n) for n = 0..1000 (terms n=1..300 from Vincenzo Librandi)
M. Albert, R. Aldred, M. Atkinson, C. Handley, D. Holton, D. McCaughan and H. van Ditmarsch, Sorting Classes, Elec. J. of Comb. 12 (2005) R31.
David Callan, A bijection for two sequences in OEIS, arXiv:1602.08347 [math.CO], 2016.
David Callan, and Toufik Mansour, Five subsets of permutations enumerated as weak sorting permutations, arXiv:1602.05182 [math.CO], 2016.
Alice L. L. Gao and Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, arXiv:1903.08946 [math.CO], 2019.
Alice L. L. Gao, and Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, The Electronic Journal of Combinatorics 26(3) (2019), P3.26.

Rest[ CoefficientList[ Series[(3 - 13x + 2x^2 + (5x - 1)*Sqrt[1 - 4x])/(2*(1 - 4x - x^2)), {x, 0, 24}], x]] (* Robert G. Wilson v, Nov 04 2005 *)

O.g.f.: (3-13*x+2*x^2+(5*x-1)*sqrt(1-4*x))/(2*(1-4*x-x^2)).
From Gary W. Adamson, Nov 14 2011: (Start)
a(n) is the sum of top row terms of M^(n-1), M is an infinite square production matrix with powers of 2 as the left border as follows:
  1, 1, 0, 0, 0, ...
  2, 1, 1, 0, 0, ...
  4, 1, 1, 1, 0, ...
  8, 1, 1, 1, 1, ...
  ... (End)
The top rows of these matrix powers, 1; 1,1; 3,2,1; 11,6,3,1; 43,21,10,4,1; appear also as columns in A026736. - R. J. Mathar, Nov 15 2011
D-finite with recurrence n*a(n) + (16-13*n)*a(n-1)+(55*n-134)*a(n-2) + (264-71*n)*a(n-3) + 10*(7-2*n)*a(n-4) = 0. - R. J. Mathar, Nov 15 2011
Shorter recurrence: n*(n+5)*a(n) = 2*(4*n^2 + 17*n - 30)*a(n-1) - 3*(5*n^2 + 17*n - 80)*a(n-2) - 2*(n+6)*(2*n-5)*a(n-3). - Vaclav Kotesovec, Oct 18 2012
a(n) ~ (5/2-11/10*sqrt(5))*(sqrt(5)+2)^n. - Vaclav Kotesovec, Oct 18 2012

More terms from Robert G. Wilson v, Nov 04 2005

a(0)=1 prepended by Alois P. Heinz, Dec 11 2020

A026673 a(n) = T(2n,n-2), T given by A026670.

Original entry on oeis.org

1, 7, 37, 177, 808, 3596, 15764, 68446, 295294, 1268356, 5430734, 23199304, 98933705, 421352919, 1792709561, 7621345733, 32380443643, 137504761035, 583684770103, 2476836131227, 10507517431481, 44566369523517, 188988331406117

Offset: 2

Clark Kimberling

nonn

Also a(n) = T(2n,n-2) = T(2n+1,n+2), T given by A026725.
Also a(n) = T(2n,n-2), T given by A026736.
Column k=6 of triangle A236830. - Philippe Deléham, Feb 02 2014

G. C. Greubel, Table of n, a(n) for n = 2..1000

Cf. A000108, A026670, A026725, A026736, A236830.

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!(  (1-Sqrt(1-4*x))^6/(8*x^2*(8*x^2 -(1-Sqrt(1-4*x))^3)) )); // G. C. Greubel, Jul 16 2019

Drop[CoefficientList[Series[(1-Sqrt[1-4*x])^6/(8*x^2*(8*x^2-(1-Sqrt[1 - 4*x])^3)), {x,0,30}], x], 2] (* G. C. Greubel, Jul 16 2019 *)

my(x='x+O('x^30)); Vec( (1-sqrt(1-4*x))^6/(8*x^2*(8*x^2 -(1-sqrt(1-4*x))^3))) \\ G. C. Greubel, Jul 16 2019

a=((1-sqrt(1-4*x))^6/(8*x^2*(8*x^2 -(1-sqrt(1-4*x))^3))).series(x, 30).coefficients(x, sparse=False); a[2:] # G. C. Greubel, Jul 16 2019

G.f.: (x^2*C(x)^6)/(1-x*C(x)^3) where C(x) is the g.f. of A000108. - Philippe Deléham, Feb 02 2014

-(n+2)*(3*n-7)*a(n) +2*(12*n^2-19*n-16)*a(n-1) +5*(-9*n^2+27*n-22)*a(n-2) -2*(3*n-4)*(2*n-3)*a(n-3)=0. - R. J. Mathar, Oct 26 2019

A026676 a(n) = T(n, floor(n/2)), T given by A026670.

Original entry on oeis.org

1, 1, 3, 4, 11, 16, 43, 65, 173, 267, 707, 1105, 2917, 4597, 12111, 19196, 50503, 80380, 211263, 337284, 885831, 1417582, 3720995, 5965622, 15652239, 25130844, 65913927, 105954110, 277822147, 447015744, 1171853635, 1886996681

Offset: 0

Clark Kimberling

nonn

Also a(n) = T(n,m) + T(n,m+1) + ... + T(n,n), m=[ (n+1)/2 ], T given by A026736.

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

Cf. A026670, A026736.

T:= function(n, k)
    if k=0 or k=n then return 1;
    elif k=n-1 then return n;
    elif (n mod 2)=0 and k=Int((n-2)/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;
List([0..20], n-> Sum([Int((n+1)/2)..n], k-> T(n, k) )); # G. C. Greubel, Jul 19 2019

T[n_, k_]:= T[n, k] = If[k==0 || k==n, 1, If[k==n-1, n, If[EvenQ[n] && k==(n-2)/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, Floor[(n+1)/2], n}], {n, 0, 40}] (* G. C. Greubel, Jul 19 2019 *)

T(n, k) = if(k==n || k==0, 1, k==n-1, n, if((n%2)==0 && k==(n-2)/2, T(n-1, k-1) + T(n-2, k-1) + T(n-1, k), T(n-1, k-1) + T(n-1, k) ));
vector(20, n, n--; sum(k=(n+1)\2, n, T(n, k)) ) \\ G. C. Greubel, Jul 19 2019

@CachedFunction
def T(n, k):
    if (k==0 or k==n): return 1
    elif (k==n-1): return n
    elif (mod(n, 2)==0 and k==(n-2)/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 (floor((n+1)/2)..n)) for n in (0..40)] # G. C. Greubel, Jul 19 2019

cp's OEIS Frontend

A027217 a(n) = Sum_{k=0..n-2} T(n,k)*T(n,k+2), T given by A026736.

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A027218 a(n) = Sum_{k=0..n-3} T(n,k)*T(n,k+3), T given by A026736.

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A027219 a(n) = Sum_{k=0..n} (k+1) * A026736(n,k).

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A027220 a(n) = Sum_{k=0..n} (k+1) * A026736(n,n-k).

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A027221 Sum of squares of numbers in row n of array T given by A026736.

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A027214 a(n) = greatest number in row n of array T given by A026736.

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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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A111279 Number of permutations avoiding the patterns {3241,3421,4321}; number of weak sorting class based on 3241.

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