A117425 Triangle T, read by rows, formed by a column bisection of triangle A117418: column k of T equals column 2*k of A117418.

1, 1, 1, 1, 3, 1, 1, 8, 5, 1, 1, 22, 20, 7, 1, 1, 65, 79, 37, 9, 1, 1, 208, 322, 180, 58, 11, 1, 1, 723, 1385, 871, 339, 83, 13, 1, 1, 2721, 6293, 4296, 1935, 550, 113, 15, 1, 1, 11053, 30152, 21821, 11092, 3465, 846, 148, 17, 1, 1, 48220, 151842, 114676, 64748, 21514, 5911, 1220, 186, 19, 1

Offset

0,5

Links

G. C. Greubel, Rows n = 0.50 of the triangle, flattened

Formula

T(n, k) = A117418(n+k, 2*k). - G. C. Greubel, May 31 2021

Example

Triangle begins:
  1;
  1,     1;
  1,     3,     1;
  1,     8,     5,     1;
  1,    22,    20,     7,     1;
  1,    65,    79,    37,     9,    1;
  1,   208,   322,   180,    58,   11,   1;
  1,   723,  1385,   871,   339,   83,  13,   1;
  1,  2721,  6293,  4296,  1935,  550, 113,  15,  1;
  1, 11053, 30152, 21821, 11092, 3465, 846, 148, 17, 1;
Column k of T equals column 2*k of A117418, which begins:
  1;
  1,   1;
  1,   2,   1;
  1,   4,   3,   1;
  1,   9,   8,   4,  1;
  1,  23,  22,  14,  5,  1;
  1,  66,  65,  50, 20,  6, 1;
  1, 209, 208, 191, 79, 28, 7, 1;
Let matrix R = SHIFT_RIGHT(A117418):
  1;
  0, 1;
  0, 1,  1;
  0, 1,  2,  1;
  0, 1,  4,  3,  1;
  0, 1,  9,  8,  4,  1;
  0, 1, 23, 22, 14,  5, 1;
  0, 1, 66, 65, 50, 20, 6, 1;
then the matrix product A117418*R yields this triangle.

Mathematica

A117418[n_, k_]:= A117418[n, k]= If[k<0 || k>n, 0, If[k==0 || k==n, 1, If[k==n-1, n, Sum[A117418[n-Floor[(k+1)/2], Floor[k/2] +j]*A117418[Floor[(k-1)/2] +j, Floor[(k-1)/2]], {j, 0, n-k}] ]]];
A117425[n_, k_]:= A117418[n+k, 2*k];
Table[A117425[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, May 31 2021 *)

Prog

(PARI)
A117418(n, k) = if(n<k || k<0, 0, if(n==k || k==0, 1, if(n==k+1, n, sum(j=0, n-k, A117418(n-((k+1)\2), k\2+j)*A117418((k-1)\2+j, (k-1)\2)))));
A117425(n, k) = A117418(n+k, 2*k);
for(n=0, 12, for(k=0, n, print1(A117425(n, k), ", "))) \\ modified by G. C. Greubel, May 31 2021
(Sage)
@CachedFunction
def A117418(n, k):
    if (k<0 or k>n): return 0
    elif (k==0 or k==n): return 1
    elif (k==n-1): return n
    else: return sum( A117418(n -(k+1)//2, k//2 +j)*A117418((k-1)//2 +j, (k-1)//2) for j in (0..n-k))
def A117425(n, k): return A117418(n+k, 2*k)
flatten([[A117425(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 31 2021

Crossrefs

Cf. A117418, A117426 (row sums), A117427 (dual).

Sequence in context: A152879 A098747 A122897 * A287215 A168216 A091698

Adjacent sequences: A117422 A117423 A117424 * A117426 A117427 A117428

Keyword

nonn,tabl

Author

Paul D. Hanna, Mar 14 2006

Status

approved