Convolution

These formulas produce table A051731 in the oeis.
http://oeis.org/A051731

Excel spreadsheet formula, American version:

=IF(COLUMN()=1, 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))

Excel spreadsheet formula, European version:

=IF(COLUMN()=1; 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))

Written differently:

T(n,1)=1, k>1: T(n,k) =  \sum\limits_{n=1}^{k-1} T(n-i,k-1) - \sum\limits_{n=1}^{k-1} T(n-i,k)

Mats Granvik (mats.granvik(AT)abo.fi)

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