A229557 Array read by antidiagonals. Rows are the denominators of consecutive harmonic transforms starting with a first row 1, 1, 1,....

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, 20, 113025, 810665, 5992, 13, 1, 1, 1, 140, 5538225, 286794631705, 5886103384, 164541, 21, 1, 1, 1, 280, 60920475, 5619905141583441965, 4630449259971272605672, 14469935305431, 1031079, 34, 1, 1, 1

Offset: 1

Author

Franz Vrabec, Sep 26 2013

Keywords

• nonn
• tabl
• frac

Comments

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).

Examples

			Table begins
1, 1,   1,      1,...
1, 1,   1,      1,...
1, 2,   6,     12,...
1, 3,  33,    825,...
1, 5, 365, 810665,...
		

Crossrefs

Cf. A229556 (numerators).

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

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

Programs

Maple

A229556A := proc(n,k)
    option remember;
    if n = 1 then
        1;
    else
        add( 1/procname(n-1,c),c=1..k) ;
    end if;
end proc:
A229557 := proc(n,k)
    denom(A229556A(n,k)) ;
end proc:
for d from 2 to 12 do
    for k from d-1 to 1 by -1 do
        printf("%d,",A229557(d-k,k)) ;
    end do:
end do: