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A118344 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. 1

%I

%S 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,

%T 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,

%U 48,27,8,1,0,1,10,44,110,165,132,90,75,35,9,1,0,1,11,54,154,275,297,132

%N 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.

%C See A118340 for definition of pendular triangles and pendular sums.

%F 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.

%e Row 6 equals the pendular sums of row 5:

%e [1, 4, 5, 3, 1, 0], where the sums proceed as follows:

%e [1,__,__,__,__,__]: T(6,0) = T(5,0) = 1;

%e [1,__,__,__,__, 1]: T(6,5) = T(6,0) - T(5,5) = 1 - 0 = 1;

%e [1, 5,__,__,__, 1]: T(6,1) = T(6,5) + T(5,1) = 1 + 4 = 5;

%e [1, 5,__,__, 4, 1]: T(6,4) = T(6,1) - T(5,4) - T(5,5) = 5-1-0 = 4;

%e [1, 5, 9,__, 4, 1]: T(6,2) = T(6,4) + T(5,2) = 4 + 5 = 9;

%e [1, 5, 9, 5, 4, 1]: T(6,3) = T(6,2) - T(5,3) - T(5,4) = 9-3-1 = 5;

%e [1, 5, 9, 5, 4, 1, 0] finally, append a zero to obtain row 6.

%e Triangle begins:

%e 1;

%e 1, 0;

%e 1, 1, 0;

%e 1, 2, 1, 0;

%e 1, 3, 2, 1, 0;

%e 1, 4, 5, 3, 1, 0;

%e 1, 5, 9, 5, 4, 1, 0;

%e 1, 6, 14, 14, 9, 5, 1, 0;

%e 1, 7, 20, 28, 14, 14, 6, 1, 0;

%e 1, 8, 27, 48, 42, 28, 20, 7, 1, 0;

%e 1, 9, 35, 75, 90, 42, 48, 27, 8, 1, 0;

%e 1, 10, 44, 110, 165, 132, 90, 75, 35, 9, 1, 0;

%e 1, 11, 54, 154, 275, 297, 132, 165, 110, 44, 10, 1, 0; ...

%e Central terms are Catalan numbers T(2*n,n) = A000108(n);

%e semi-diagonals form successive self-convolutions of the central terms:

%e T(2*n+1,n) = [A000108^2](n),

%e T(2*n+2,n) = [A000108^3](n).

%o (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)) ))))

%Y Cf. A000108, A118340, A033184.

%K nonn,tabl

%O 0,8

%A _Paul D. Hanna_, Apr 26 2006

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Last modified January 24 23:49 EST 2021. Contains 340414 sequences. (Running on oeis4.)