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A096811
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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).
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5
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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
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OFFSET
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0,19
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COMMENTS
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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.
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LINKS
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FORMULA
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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.
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EXAMPLE
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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,...].
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PROG
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(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(""))
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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