A096815 Triangle, read by rows, such that T(n,k) equals the k-th term of the convolution of the two prior rows indexed by (n-k) and k.

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 3, 1, 1, 1, 4, 4, 4, 1, 1, 1, 5, 6, 7, 5, 1, 1, 1, 6, 9, 11, 9, 6, 1, 1, 1, 7, 11, 17, 17, 13, 7, 1, 1, 1, 8, 14, 24, 30, 26, 16, 8, 1, 1, 1, 9, 18, 32, 42, 50, 36, 21, 9, 1, 1, 1, 10, 21, 42, 61, 79, 76, 51, 25, 10, 1, 1, 1, 11, 25, 53, 85, 118, 129, 115, 67

Offset

0,9

Comments

Row sums are A096816.

Links

Table of n, a(n) for n=0..87.

Formula

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

Example

T(7,4) = 11 = 4th term of the convolution of row (7-4) and row 4:
T(3,0)*T(4,4) + T(3,1)*T(4,3) + T(3,2)*T(4,2) + T(3,3)*T(4,1).
Rows begin:
[1],
[1,1],
[1,1,1],
[1,1,2,1],
[1,1,3,3,1],
[1,1,4,4,4,1],
[1,1,5,6,7,5,1],
[1,1,6,9,11,9,6,1],
[1,1,7,11,17,17,13,7,1],
[1,1,8,14,24,30,26,16,8,1],
[1,1,9,18,32,42,50,36,21,9,1],
[1,1,10,21,42,61,79,76,51,25,10,1],
[1,1,11,25,53,85,118,129,115,67,31,11,1],...

Prog

(PARI) T(n, k)=if(n<k || k<0, 0, if(k<=1 || k==n, 1, sum(j=0, k, T(n-k, j)*T(k, k-j))))
(Maxima) T(n, k):= if ( n<k or k<0 ) then 0 else
if ( k<=1 or k=n ) then 1 else sum(T(n-k, j)*T(k, k-j), j, 0, k);
create_list(T(n, k), n, 0, 12, k, 0, n); /* Emanuele Munarini, May 12 2012 */

Crossrefs

Cf. A096816, A096811.

Sequence in context: A136043 A336420 A254055 * A193516 A124445 A124279

Adjacent sequences: A096812 A096813 A096814 * A096816 A096817 A096818

Keyword

nonn,tabl

Author

Paul D. Hanna, Jul 20 2004

Status

approved