A115085 - OEIS

A115085

Triangle, read by rows, where T(n,k) equals the dot product of the vector of terms in row n-1 from T(n-1,k) to T(n-1,n-1) with the vector of terms in column k+1 from T(k+1,k+1) to T(n,k+1): T(n,k) = Sum_{j=0..n-k-1} T(n-1,j+k)*T(j+k+1,k+1) for n>k+1>0, with T(n,n) = 1 and T(n,n-1) = n (n>=1).

1, 1, 1, 3, 2, 1, 12, 5, 3, 1, 58, 21, 7, 4, 1, 321, 102, 32, 9, 5, 1, 1963, 579, 158, 45, 11, 6, 1, 13053, 3601, 933, 226, 60, 13, 7, 1, 92946, 24426, 5939, 1395, 306, 77, 15, 8, 1, 702864, 176858, 41385, 9097, 1977, 398, 96, 17, 9, 1, 5599204, 1359906, 306070

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,4

COMMENTS

Triangle A115080 is the dual of this triangle.

LINKS

Paul D. Hanna, Table of n, a(n) for n = 0..405, as a flattened triangle of rows 0..27.

EXAMPLE

T(n,k)=[T(n-1,k),T(n-1,k+1),..,T(n-1,n-1)]*[T(k+1,k+1),T(k+2,k+1),..,T(n,k+1)]:

12 = [3,2,1]*[1,2,5] = 3*1 + 2*2 + 1*5;

21 = [5,3,1]*[1,3,7] = 5*1 + 3*3 + 1*7;

102 = [21,7,4,1]*[1,3,7,32] = 21*1 + 7*3 + 4*7 + 1*32;

158 = [32,9,5,1]*[1,4,9,45] = 32*1 + 9*4 + 5*9 + 1*45.

Triangle begins:

1, 1;

3, 2, 1;

12, 5, 3, 1;

58, 21, 7, 4, 1;

321, 102, 32, 9, 5, 1;

1963, 579, 158, 45, 11, 6, 1;

13053, 3601, 933, 226, 60, 13, 7, 1;

92946, 24426, 5939, 1395, 306, 77, 15, 8, 1;

702864, 176858, 41385, 9097, 1977, 398, 96, 17, 9, 1;

5599204, 1359906, 306070, 65310, 13195, 2691, 502, 117, 19, 10, 1;

46746501, 10996740, 2403792, 494022, 97701, 18353, 3549, 618, 140, 21, 11, 1;

407019340, 93136545, 19799468, 3970878, 755834, 140178, 24691, 4563, 746, 165, 23, 12, 1; ...

PROG

(PARI) {T(n, k)=if(n==k, 1, if(n==k+1, n, sum(j=0, n-k-1, T(n-1, j+k)*T(j+k+1, k+1))))}

for(n=0, 12, for(k=0, n, print1(T(n, k), ", ")); print(""))

CROSSREFS

Cf. A115086 (column 0), A115087 (column 1), A115088 (column 2), A115089 (row sums); A115080 (dual triangle).

Sequence in context: A123513 A117442 A118435 * A110616 A059418 A092582

Adjacent sequences: A115082 A115083 A115084 * A115086 A115087 A115088

KEYWORD

nonn,tabl

AUTHOR

Paul D. Hanna, Jan 13 2006

STATUS

approved