A026769 Triangular array T read by rows: T(n,0)=T(n,n)=1 for n >= 0; T(2,1)=2; for n >= 3 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<=(n-1)/2, else T(n,k) = T(n-1,k-1) + T(n-1,k).

1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 6, 7, 4, 1, 1, 8, 17, 11, 5, 1, 1, 10, 31, 28, 16, 6, 1, 1, 12, 49, 76, 44, 22, 7, 1, 1, 14, 71, 156, 120, 66, 29, 8, 1, 1, 16, 97, 276, 352, 186, 95, 37, 9, 1, 1, 18, 127, 444, 784, 538, 281, 132, 46, 10, 1, 1, 20, 161, 668, 1504, 1674, 819, 413, 178, 56, 11, 1

Offset: 0

Author

Clark Kimberling

Keywords

• nonn
• tabl

Comments

T(n, k) is the number of paths from (0, 0) to (k,n-k) in the directed graph having vertices (i, j) (i and j in range [0,n]) 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>=1.

Also, square array R read by antidiagonals where R(i,j) = T(i+j,i), which is equal to the number of paths from (0,0) to (i,j) in the above graph. - Max Alekseyev, Dec 02 2015

Examples

			Triangle begins as:
  1;
  1,  1;
  1,  2,   1;
  1,  4,   3,   1;
  1,  6,   7,   4,   1;
  1,  8,  17,  11,   5,   1;
  1, 10,  31,  28,  16,   6,   1;
  1, 12,  49,  76,  44,  22,   7,   1;
  1, 14,  71, 156, 120,  66,  29,   8,  1;
  1, 16,  97, 276, 352, 186,  95,  37,  9,  1;
  1, 18, 127, 444, 784, 538, 281, 132, 46, 10, 1;
		

Links

• 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.

Crossrefs

Cf. A026770, A026771, A026772, A026773, A026774, A026775, A026776, A026777, A026779

Cf. A026780 (a variant with h>=0)

Programs

GAP

T:= function(n,k)
    if k=0 or k=n then return 1;
    elif (n=2 and k=1) then return 2;
    elif (k <= Int((n-1)/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

Maple

A026769 := proc(n,k)
    option remember;
    if k= 0 or k =n then
        1;
    elif n= 2 and k= 1 then
        2;
    elif k <= (n-1)/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: # R. J. Mathar, Jun 15 2014

Mathematica

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]];
Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 10 2017, from Maple *)

PARI

T(n,k) = if(k==0 || k==n, 1, if(n==2 && k==1, 2, if( k<=(n-1)/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

Sage

@CachedFunction
def T(n, k):
    if (k==0 or k==n): return 1
    elif (n==2 and k==1): return 2
    elif (k<=(n-1)/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

Formula

For n>=2*k, T(n,k) = coefficient of x^k in G(x)*S(x)^(n-2*k). For n<=2*k, T(n,k) = coefficient of x^(n-k) in G(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 G(x)=1/(1-x*(C(x)+S(x))) is o.g.f. for A026770. - Max Alekseyev, Dec 02 2015

Extensions

Offset corrected by R. J. Mathar, Jun 15 2014

More terms added by G. C. Greubel, Oct 31 2019