A026747 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 n is even and 1 <= k <= n/2, else T(n,k) = T(n-1,k-1) + T(n-1,k).

1, 1, 1, 1, 3, 1, 1, 4, 4, 1, 1, 6, 11, 5, 1, 1, 7, 17, 16, 6, 1, 1, 9, 30, 44, 22, 7, 1, 1, 10, 39, 74, 66, 29, 8, 1, 1, 12, 58, 143, 184, 95, 37, 9, 1, 1, 13, 70, 201, 327, 279, 132, 46, 10, 1, 1, 15, 95, 329, 671, 790, 411, 178, 56, 11, 1, 1, 16, 110, 424, 1000, 1461, 1201, 589, 234, 67, 12, 1

Offset

0,5

Links

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Formula

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

Example

Triangle begins as:
  1;
  1, 1;
  1, 3,  1;
  1, 4,  4,  1;
  1, 6, 11,  5,  1;
  1, 7, 17, 16,  6, 1;
  1, 9, 30, 44, 22, 7, 1;

Maple

A026747 := proc(n, k)
    if k=0 or k = n then
        1;
    elif type(n, 'even') and 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) ;
    end if ;
end proc: # R. J. Mathar, Jun 30 2013

Mathematica

T[n_, k_]:= T[n, k]= If[k==0 || k==n, 1, If[EvenQ[n] && 1<=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, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Oct 28 2019 *)

Prog

(PARI) T(n, k) = if(k==0 || k==n, 1, if(n%2==0 && 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 28 2019
(Sage)
def T(n, k):
    if (k==0 or k==n): return 1
    elif (mod(n, 2)==0 and 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 28 2019
(GAP)
T:= function(n, k)
    if k=0 or k=n then return 1;
    elif (n mod 2)=0 and k<Int(n/2)+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..12], n-> List([0..n], k-> T(n, k) ))); # G. C. Greubel, Oct 28 2019

Crossrefs

Cf. A026754 (row sums).

Sequence in context: A026626 A136482 A026648 * A026374 A174032 A180979

Adjacent sequences: A026744 A026745 A026746 * A026748 A026749 A026750

Keyword

nonn,tabl

Author

Clark Kimberling

Extensions

More terms added by G. C. Greubel, Oct 28 2019

Status

approved