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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).
5
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
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
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, Jul 20 2004
STATUS
approved