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A118344
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Pendular Catalan triangle, read by rows, where row n is formed from row n-1 by the recurrence: if n > 2k, T(n,k) = T(n,n-k) + T(n-1,k), else T(n,k) = T(n,n-1-k) - T(n-1,k) - T(n-1,k+1), for n>=k>=0, with T(n,0)=1 and T(n,n)=0^n.
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1
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1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 2, 1, 0, 1, 4, 5, 3, 1, 0, 1, 5, 9, 5, 4, 1, 0, 1, 6, 14, 14, 9, 5, 1, 0, 1, 7, 20, 28, 14, 14, 6, 1, 0, 1, 8, 27, 48, 42, 28, 20, 7, 1, 0, 1, 9, 35, 75, 90, 42, 48, 27, 8, 1, 0, 1, 10, 44, 110, 165, 132, 90, 75, 35, 9, 1, 0, 1, 11, 54, 154, 275, 297, 132
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OFFSET
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0,8
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COMMENTS
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See A118340 for definition of pendular triangles and pendular sums.
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LINKS
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Table of n, a(n) for n=0..84.
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FORMULA
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T(2*n+m,n) = [A000108^(m+1)](n), i.e., the m-th lower semi-diagonal forms the self-convolution (m+1)-power of A000108.
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EXAMPLE
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Row 6 equals the pendular sums of row 5:
[1, 4, 5, 3, 1, 0], where the sums proceed as follows:
[1,__,__,__,__,__]: T(6,0) = T(5,0) = 1;
[1,__,__,__,__, 1]: T(6,5) = T(6,0) - T(5,5) = 1 - 0 = 1;
[1, 5,__,__,__, 1]: T(6,1) = T(6,5) + T(5,1) = 1 + 4 = 5;
[1, 5,__,__, 4, 1]: T(6,4) = T(6,1) - T(5,4) - T(5,5) = 5-1-0 = 4;
[1, 5, 9,__, 4, 1]: T(6,2) = T(6,4) + T(5,2) = 4 + 5 = 9;
[1, 5, 9, 5, 4, 1]: T(6,3) = T(6,2) - T(5,3) - T(5,4) = 9-3-1 = 5;
[1, 5, 9, 5, 4, 1, 0] finally, append a zero to obtain row 6.
Triangle begins:
1;
1, 0;
1, 1, 0;
1, 2, 1, 0;
1, 3, 2, 1, 0;
1, 4, 5, 3, 1, 0;
1, 5, 9, 5, 4, 1, 0;
1, 6, 14, 14, 9, 5, 1, 0;
1, 7, 20, 28, 14, 14, 6, 1, 0;
1, 8, 27, 48, 42, 28, 20, 7, 1, 0;
1, 9, 35, 75, 90, 42, 48, 27, 8, 1, 0;
1, 10, 44, 110, 165, 132, 90, 75, 35, 9, 1, 0;
1, 11, 54, 154, 275, 297, 132, 165, 110, 44, 10, 1, 0; ...
Central terms are Catalan numbers T(2*n,n) = A000108(n);
semi-diagonals form successive self-convolutions of the central terms:
T(2*n+1,n) = [A000108^2](n),
T(2*n+2,n) = [A000108^3](n).
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PROG
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(PARI) {T(n, k)=if(n<k|k<0, 0, if(k==0, 1, if(n==k, 0, if(n>2*k, T(n, n-k)+T(n-1, k), T(n, n-1-k)-T(n-1, k)-if(n-1>k, T(n-1, k+1)) ))))}
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CROSSREFS
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Cf. A000108, A118340.
Cf. A033184 [From Philippe DELEHAM, Nov 13 2009]
Sequence in context: A100995 A136566 A048983 * A119270 A175804 A195017
Adjacent sequences: A118341 A118342 A118343 * A118345 A118346 A118347
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KEYWORD
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nonn,tabl
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AUTHOR
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Paul D. Hanna, Apr 26 2006
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STATUS
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approved
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