A128320 Triangle, read by rows, where T(n,k) equals the dot product of the vector of terms in row n that are to the right of T(n,k) with the vector of terms in column k that are above T(n,k) for n>k+1>0, with the odd numbers in the secondary diagonal and all 1's in the main diagonal.

1, 1, 1, 4, 3, 1, 17, 8, 5, 1, 98, 41, 12, 7, 1, 622, 234, 73, 16, 9, 1, 4512, 1602, 418, 113, 20, 11, 1, 35373, 11976, 3110, 650, 161, 24, 13, 1, 300974, 98541, 23920, 5242, 930, 217, 28, 15, 1, 2722070, 866942, 207549, 41304, 8094, 1258, 281, 32, 17, 1

Offset: 0

Author

Paul D. Hanna, Feb 25 2007

Keywords

• nonn
• tabl

Examples

			Illustrate the recurrence by:
  T(n,k) = [T(n,k+1),T(n,k+2), ..,T(n,n)]*[T(k,k),T(k+1,k),..,T(n-1,k)]:
  T(3,0) = [8,5,1]*[1,1,4]~ = 8*1 + 5*1 + 1*4 = 17;
  T(4,1) = [12,7,1]*[1,3,8]~ = 12*1 + 7*3 + 1*8 = 41;
  T(5,1) = [73,16,9,1]*[1,3,8,41]~ = 73*1 + 16*3 + 9*8 + 1*41 = 234;
  T(6,2) = [113,20,11,1]*[1,5,12,73]~ = 113*1 + 20*5 + 11*12 + 1*73 = 418.
Triangle begins:
         1;
         1,       1;
         4,       3,       1;
        17,       8,       5,      1;
        98,      41,      12,      7,     1;
       622,     234,      73,     16,     9,     1;
      4512,    1602,     418,    113,    20,    11,    1;
     35373,   11976,    3110,    650,   161,    24,   13,   1;
    300974,   98541,   23920,   5242,   930,   217,   28,  15,  1;
   2722070,  866942,  207549,  41304,  8094,  1258,  281,  32, 17,  1;
  26118056, 8139602, 1885166, 377757, 65088, 11762, 1634, 353, 36, 19, 1;
		

Links

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

Crossrefs

Columns k: A128321 (k=0), A128322 (k=1), A128323 (k=2).

Sums: A128324 (row sums).

Variant of: A115080.

Programs

Magma

function T(n,k) // T = A128320
   if k eq n then return 1;
   elif k eq n-1 then return 2*n-1;
   else return (&+[T(n, k+j+1)*T(k+j, k): j in [0..n-k-1]]);
   end if;
end function;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 25 2024

Mathematica

T[n_, k_]:= T[n, k]= If[k==n, 1, If[k==n-1, 2*n-1, Sum[T[n,k+j+1] *T[k+j,k], {j,0,n-k-1}]]];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jun 25 2024 *)

PARI

{T(n,k)=if(n==k,1, if(n==k+1,2*n-1, sum(i=0,n-k-1, T(n,k+i+1)*T(k+i,k))))};
for(n=0, 12, for(k=0, n, print1(T(n, k), ", ")); print(""))

SageMath

@CachedFunction
def T(n,k): # T = A128320
    if k==n: return 1
    elif k==n-1: return 2*n-1
    else: return sum(T(n, k+j+1)*T(k+j, k) for j in range(n-k))
flatten([[T(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jun 25 2024

Formula

T(n,k) = Sum_{j=0..n-1-k} T(n,k+j+1)*T(k+j,k) for n > k+1 > 0, with T(n,n) = 1 and T(n, n-1) = 2*n-1 for k >= 0.