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A179285
Triangle T(n,k) read by rows, defined by: T(1,1)=1; n > 1 and k=1: T(n,1) = T(n-1,2) + T(n,2); k=2: T(n,2) = A000196(n-1); k > 2: T(n,k) = (Sum_{i=1..k-1} T(n-i,k-1)) - (Sum_{i=1..k-1} T(n-i,k)).
2
1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 3, 2, 0, 1, 1, 4, 2, 2, 0, 1, 1, 4, 2, 2, 1, 0, 1, 1, 4, 2, 0, 2, 1, 0, 1, 1, 4, 2, 2, 1, 1, 1, 0, 1, 1, 5, 3, 2, 0, 1, 1, 1, 0, 1, 1, 6, 3, 1, 1, 1, 0, 1, 1, 0, 1, 1, 6, 3, 3, 3, 0, 1, 0, 1, 1, 0, 1, 1, 6, 3, 2, 2, 2, 1, 0, 0, 1, 1, 0, 1, 1, 6, 3, 1, 0, 2, 1, 1, 0, 0, 1, 1, 0, 1, 1
OFFSET
1,4
COMMENTS
The second column, sequence A000196, is the initial condition for the recurrence in this triangle. See A051731, formula entered on Feb 16 2010 for the more pure form of this recurrence.
FORMULA
T(1,1)=1; n > 1 and k=1: T(n,1) = T(n-1,2) + T(n,2); k=2: T(n,2) = A000196(n-1); k > 2: T(n,k) = (Sum_{i=1..k-1} T(n-i,k-1)) - (Sum_{i=1..k-1} T(n-i,k)).
EXAMPLE
Triangle begins:
1;
1, 1;
2, 1, 1;
2, 1, 1, 1;
3, 2, 0, 1, 1;
4, 2, 2, 0, 1, 1;
4, 2, 2, 1, 0, 1, 1;
4, 2, 0, 2, 1, 0, 1, 1;
4, 2, 2, 1, 1, 1, 0, 1, 1;
5, 3, 2, 0, 1, 1, 1, 0, 1, 1;
6, 3, 1, 1, 1, 0, 1, 1, 0, 1, 1;
PROG
(Excel) Using European dot comma style:
=if(and(row()=1; column()=1); 1; if(row()>=column(); if(column()=1; indirect(address(row()-1; column()+1))+indirect(address(row(); column()+1)); if(column()=2; floor(((row()-1)^0, 5); 1); if(row()>=column(); sum(indirect(address(row()-column()+1; column()-1; 4)&":"&address(row()-1; column()-1; 4); 4))-sum(indirect(address(row()-column()+1; column(); 4)&":"&address(row()-1; column(); 4); 4)); 0))); 0))
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Mats Granvik, Jul 09 2010
STATUS
approved