login
Array read by antidiagonals. Rows are the denominators of consecutive harmonic transforms starting with a first row 1, 1, 1,....
2

%I #14 Oct 06 2013 09:26:48

%S 1,1,1,1,1,1,1,1,2,1,1,1,6,3,1,1,1,12,33,5,1,1,1,60,825,365,8,1,1,1,

%T 20,113025,810665,5992,13,1,1,1,140,5538225,286794631705,5886103384,

%U 164541,21,1,1,1,280,60920475,5619905141583441965,4630449259971272605672,14469935305431,1031079,34,1,1,1

%N Array read by antidiagonals. Rows are the denominators of consecutive harmonic transforms starting with a first row 1, 1, 1,....

%C The "harmonic transform" of a sequence of positive numbers a(i) is the sequence h(n) of the partial sums of their reciprocals: h(n)=sum_{i=1..n} 1/a(i).

%e Table begins

%e 1, 1, 1, 1,...

%e 1, 1, 1, 1,...

%e 1, 2, 6, 12,...

%e 1, 3, 33, 825,...

%e 1, 5, 365, 810665,...

%p A229556A := proc(n,k)

%p option remember;

%p if n = 1 then

%p 1;

%p else

%p add( 1/procname(n-1,c),c=1..k) ;

%p end if;

%p end proc:

%p A229557 := proc(n,k)

%p denom(A229556A(n,k)) ;

%p end proc:

%p for d from 2 to 12 do

%p for k from d-1 to 1 by -1 do

%p printf("%d,",A229557(d-k,k)) ;

%p end do:

%p end do:

%Y Cf. A229556 (numerators).

%Y Rows 1-4 are A000012(n), A000012(n), A002805(n), A124432(n+1).

%Y Columns 1-2 are A000012(n), A000045(n+1).

%K nonn,tabl,frac

%O 1,9

%A _Franz Vrabec_, Sep 26 2013