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

0,4

Links

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

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.

Example

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;

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 *)

Prog

(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(""))
(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
(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

Crossrefs

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

Sums: A128324 (row sums).

Variant of: A115080.

Sequence in context: A098234 A193795 A181355 * A189507 A348436 A350528

Adjacent sequences: A128317 A128318 A128319 * A128321 A128322 A128323

Keyword

nonn,tabl

Author

Paul D. Hanna, Feb 25 2007

Status

approved