A026725 Triangular array, T, read by rows: T(n,0) = T(n,n) = 1. 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 n is odd and k=n/2, otherwise T(n,k) = T(n-1,k-1) + T(n-1,k).

1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 5, 7, 4, 1, 1, 6, 16, 11, 5, 1, 1, 7, 22, 27, 16, 6, 1, 1, 8, 29, 65, 43, 22, 7, 1, 1, 9, 37, 94, 108, 65, 29, 8, 1, 1, 10, 46, 131, 267, 173, 94, 37, 9, 1, 1, 11, 56, 177, 398, 440, 267, 131, 46, 10, 1, 1, 12, 67, 233

Offset: 0

Author

Clark Kimberling

Keywords

• nonn
• tabl

Comments

T(n+2,n) = A134869(n+1). - Philippe Deléham, Feb 01 2014

Examples

			Triangle begins:
1
1  1
1  2  1
1  4  3   1
1  5  7   4   1
1  6 16  11   5    1
1  7 22  27  16    6   1
1  8 29  65  43   22   7   1
1  9 37  94 108   65  29   8   1
1 10 46 131 267  173  94  37   9  1
1 11 56 177 398  440 267 131  46 10  1
1 12 67 233 575 1105 707 398 177 56 11 1
... - _Philippe Deléham_, Feb 01 2014
		

Links

• G. C. Greubel, Rows n = 0..99 of triangle, flattened
• Rob Arthan, Comments on A026674, A026725, A026670

Crossrefs

Cf. A026674.

Programs

GAP

T:= function(n,k)
    if k=0 or k=n then return 1;
    elif 2*k=n-1 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..14], n-> List([0..n], k-> T(n,k) ))); # G. C. Greubel, Jul 16 2019

Maple

A026725 := proc(n,k)
    option remember;
    if n < 0 or k < 0 then
        0;
    elif k=0 or k=n then
        1;
    elif 2*k = n-1 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) ;
    end if;
end proc: # R. J. Mathar, Oct 21 2019

Mathematica

T[n_, k_]:= T[n, k]= If[k==0||k==n, 1, If[OddQ[n] && 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]]];
Table[T[n, k], {n,0,14}, {k,0,n}]//Flatten (* G. C. Greubel, Jul 16 2019 *)

PARI

T(n,k) = if(k==n || k==0, 1, if(2*k==n-1, 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=0,11, for(k=0,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Jul 16 2019

Sage

@CachedFunction
def T(n, k):
    if (k==0 or k==n): return 1
    elif (mod(n,2)==0 and 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..14)] # G. C. Greubel, Jul 16 2019

Formula

T(n, k) = number of paths from (0, 0) to (n-k, k) in 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+1)-to-(i+1, i+2) for i >= 0.

Comment from Rick L. Shepherd, Aug 05 2002: Probably this should be changed to "and edges (i+1, i)-to-(i+2, i+1) for i >= 0."

Extensions

Title and offset corrected by G. C. Greubel, Jul 16 2019, again by R. J. Mathar, Oct 21 2019, again by Sean A. Irvine, Oct 25 2019