A117427 Triangle T, read by rows, equal to the matrix square of triangle A117418; also equals a column bisection of triangle A117418: column 2k+1 of T^(1/2) equals column k of T.

1, 2, 1, 4, 4, 1, 9, 14, 6, 1, 23, 50, 28, 8, 1, 66, 191, 126, 48, 10, 1, 209, 780, 572, 264, 70, 12, 1, 724, 3415, 2692, 1440, 438, 98, 14, 1, 2722, 15924, 13203, 7990, 2646, 698, 130, 16, 1, 11054, 78595, 67542, 45566, 15920, 4702, 1029, 168, 18, 1, 48221, 408337, 360117, 268072, 96848, 31030, 7536, 1472, 206, 20, 1

Offset

0,2

Links

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Formula

T(n, k) = A117418(n+k+1, 2*k+1). - G. C. Greubel, May 31 2021

Example

Triangle T begins:
      1;
      2,     1;
      4,     4,     1;
      9,    14,     6,     1;
     23,    50,    28,     8,     1;
     66,   191,   126,    48,    10,    1;
    209,   780,   572,   264,    70,   12,    1;
    724,  3415,  2692,  1440,   438,   98,   14,   1;
   2722, 15924, 13203,  7990,  2646,  698,  130,  16,  1;
  11054, 78595, 67542, 45566, 15920, 4702, 1029, 168, 18, 1;
The matrix square-root T^(1/2) = 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;
where column 2k+1 of T^(1/2) equals column k of T for k>=0.

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}] ]]];
A117427[n_, k_]:= A117418[n+k+1, 2*k+1];
Table[A117427[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, May 31 2021 *)

Prog

(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 A117427(n, k): return A117418(n+k+1, 2*k+1)
flatten([[A117427(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 31 2021

Crossrefs

Cf. A117418 (T^(1/2)), A117425 (dual), A117428 (row sums).

Sequence in context: A263989 A202710 A200965 * A097761 A200756 A108556

Adjacent sequences: A117424 A117425 A117426 * A117428 A117429 A117430

Keyword

nonn,tabl

Author

Paul D. Hanna, Mar 14 2006

Status

approved