A115080 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): T(n,k) = Sum_{j=0..n-k-1} T(n,j+k+1)*T(j+k,k) for n > k+1 > 0, with T(n,n) = 1 and T(n,n-1) = n (n>=1).

1, 1, 1, 3, 2, 1, 11, 5, 3, 1, 50, 20, 7, 4, 1, 257, 94, 31, 9, 5, 1, 1467, 507, 150, 44, 11, 6, 1, 9081, 3009, 853, 218, 59, 13, 7, 1, 60272, 19350, 5251, 1307, 298, 76, 15, 8, 1, 424514, 132920, 35109, 8313, 1881, 390, 95, 17, 9, 1, 3151226, 966962, 249332

Offset

0,4

Comments

Triangle A115085 is the dual of this triangle.

Links

Paul D. Hanna, Table of n, a(n) for n = 0..350, as a flattened triangle of rows 0..25.

Example

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)]:
  11 = [5,3,1]*[1,1,3] = 5*1 + 3*1 + 1*3;
  20 = [7,4,1]*[1,2,5] = 7*1 + 4*2 + 1*5;
  94 = [31,9,5,1]*[1,2,5,20] = 31*1 + 9*2 + 5*5 + 1*20;
  150 = [44,11,6,1]*[1,3,7,31] = 44*1 + 11*3 + 6*7 + 1*31.
Triangle begins:
         1;
         1,       1;
         3,       2,       1;
        11,       5,       3,      1;
        50,      20,       7,      4,     1;
       257,      94,      31,      9,     5,     1;
      1467,     507,     150,     44,    11,     6,    1;
      9081,    3009,     853,    218,    59,    13,    7,   1;
     60272,   19350,    5251,   1307,   298,    76,   15,   8,   1;
    424514,  132920,   35109,   8313,  1881,   390,   95,  17,   9,  1;
   3151226,  966962,  249332,  57738, 12315,  2587,  494, 116,  19, 10,  1;
  24510411, 7396366, 1873214, 422948, 88737, 17377, 3437, 610, 139, 21, 11, 1;
  ...

Prog

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

Crossrefs

Cf. A115081 (column 0), A115082 (column 1), A115083 (column 2), A115084 (row sums); A115085 (dual triangle).

Sequence in context: A325305 A309951 A077756 * A222730 A104219 A123513

Adjacent sequences: A115077 A115078 A115079 * A115081 A115082 A115083

Keyword

nonn,tabl

Author

Paul D. Hanna, Jan 13 2006

Status

approved