

A118344


Pendular Catalan triangle, read by rows, where row n is formed from row n1 by the recurrence: if n > 2k, T(n,k) = T(n,nk) + T(n1,k), else T(n,k) = T(n,n1k)  T(n1,k)  T(n1,k+1), for n>=k>=0, with T(n,0)=1 and T(n,n)=0^n.


1



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
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

0,8


COMMENTS

See A118340 for definition of pendular triangles and pendular sums.


LINKS

Table of n, a(n) for n=0..84.


FORMULA

T(2*n+m,n) = [A000108^(m+1)](n), i.e., the mth lower semidiagonal forms the selfconvolution (m+1)power of A000108.


EXAMPLE

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) = 510 = 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) = 931 = 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);
semidiagonals form successive selfconvolutions of the central terms:
T(2*n+1,n) = [A000108^2](n),
T(2*n+2,n) = [A000108^3](n).


PROG

(PARI) T(n, k)=if(n<k  k<0, 0, if(k==0, 1, if(n==k, 0, if(n>2*k, T(n, nk)+T(n1, k), T(n, n1k)T(n1, k)if(n1>k, T(n1, k+1)) ))))


CROSSREFS

Cf. A000108, A118340, A033184.
Sequence in context: A321391 A244003 A332670 * A119270 A267109 A175804
Adjacent sequences: A118341 A118342 A118343 * A118345 A118346 A118347


KEYWORD

nonn,tabl


AUTHOR

Paul D. Hanna, Apr 26 2006


STATUS

approved



