A096811 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-2).

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 3, 2, 1, 1, 1, 1, 1, 2, 3, 3, 2, 1, 1, 1, 1, 1, 2, 4, 4, 3, 2, 1, 1, 1, 1, 1, 2, 4, 5, 4, 4, 2, 1, 1, 1, 1, 1, 2, 4, 6, 6, 5, 4, 2, 1, 1, 1, 1, 1, 2, 4, 7, 7, 8, 6, 4, 2, 1, 1, 1, 1, 1, 2, 4, 7, 9, 10, 9, 7, 4, 2, 1, 1, 1, 1, 1, 2, 4, 8, 10, 12, 12, 11, 7, 4, 2, 1, 1, 1, 1, 1, 2, 4, 8, 12, 14, 16, 15, 12, 7, 4, 2, 1, 1, 1, 1, 1, 2, 4, 8, 13, 17, 18, 21, 17, 13, 8, 4, 2, 1, 1, 1, 1, 1, 2, 4, 8, 14, 19, 23, 25, 24, 20, 14, 8, 4, 2, 1, 1, 1, 1, 1, 2, 4, 8, 15, 22, 27, 32, 30, 29, 23, 15, 8, 4, 2, 1, 1

Offset

0,19

Comments

Two row convergents exist simultaneously. When the rows are read forwards, they converge to A096812. When the rows are read backwards, they converge to A096813. The row sums form A096814.

Links

Paul D. Hanna, Table of n, a(n) for n = 0..5150

Formula

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

Example

T(11,5) = 6 = 5th term of convolution of row (11-5) with row (5-2) =
T(6,1)*T(3,3) + T(6,2)*T(3,2) + T(6,3)*T(3,1) + T(6,4)*T(3,0).
Rows begin with n=0:
1;
1, 1;
1, 1, 1;
1, 1, 1, 1;
1, 1, 1, 1, 1;
1, 1, 1, 2, 1, 1;
1, 1, 1, 2, 2, 1, 1;
1, 1, 1, 2, 3, 2, 1, 1;
1, 1, 1, 2, 3, 3, 2, 1, 1;
1, 1, 1, 2, 4, 4, 3, 2, 1, 1;
1, 1, 1, 2, 4, 5, 4, 4, 2, 1, 1;
1, 1, 1, 2, 4, 6, 6, 5, 4, 2, 1, 1;
1, 1, 1, 2, 4, 7, 7, 8, 6, 4, 2, 1, 1;
1, 1, 1, 2, 4, 7, 9, 10, 9, 7, 4, 2, 1, 1;
1, 1, 1, 2, 4, 8, 10, 12, 12, 11, 7, 4, 2, 1, 1;
1, 1, 1, 2, 4, 8, 12, 14, 16, 15, 12, 7, 4, 2, 1, 1;
1, 1, 1, 2, 4, 8, 13, 17, 18, 21, 17, 13, 8, 4, 2, 1, 1;
1, 1, 1, 2, 4, 8, 14, 19, 23, 25, 24, 20, 14, 8, 4, 2, 1, 1;
1, 1, 1, 2, 4, 8, 15, 22, 27, 32, 30, 29, 23, 15, 8, 4, 2, 1, 1;
1, 1, 1, 2, 4, 8, 15, 24, 31, 38, 40, 38, 35, 25, 16, 8, 4, 2, 1, 1;
1, 1, 1, 2, 4, 8, 16, 26, 36, 45, 48, 52, 46, 40, 28, 17, 8, 4, 2, 1, 1;
1, 1, 1, 2, 4, 8, 16, 28, 40, 53, 59, 66, 64, 55, 45, 30, 17, 8, 4, 2, 1, 1;
1, 1, 1, 2, 4, 8, 16, 30, 44, 60, 71, 83, 84, 78, 66, 51, 32, 17, 8, 4, 2, 1, 1;
1, 1, 1, 2, 4, 8, 16, 31, 48, 68, 83, 102, 108, 106, 95, 76, 55, 33, 18, 8, 4, 2, 1, 1; ...
Forwards row convergent forms A096812:
[1,1,1,2,4,8,16,34,72,156,336,746,1652,3696,...].
Backwards row convergent forms A096813:
[0,1,1,2,4,8,18,40,92,210,490,1178,2834,6908,...].

Prog

(PARI) {T(n, k) = if(n<k || k<0, 0, if(k<=1 || k==n, 1, sum(j=1, k-1, T(n-k, j)*T(k-2, k-j-1))))}
for(n=0, 20, for(k=0, n, print1(T(n, k), ", ")); print(""))

Crossrefs

Cf. A096813, A096814, A091499.

Sequence in context: A270992 A117546 A274196 * A082478 A279060 A324119

Adjacent sequences: A096808 A096809 A096810 * A096812 A096813 A096814

Keyword

nonn,tabl

Author

Paul D. Hanna, Jul 20 2004

Status

approved