A229556 -id:A229556 - cp's OEIS frontend

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

Franz Vrabec, Sep 26 2013

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).

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

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).

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:

A350834 Number of ways to tile an n X n right triangle with squares and dominoes, where vertical dominoes are only allowed in the largest vertical column.

Original entry on oeis.org

1, 1, 3, 11, 73, 749, 12657, 343693, 15140923, 1078147567, 124268659473, 23172219304577, 6991754237772409, 3413365649747365697, 2696315730346059254139, 3446235324323962173174283, 7127008624714819485698797681, 23848280807640171362927751869341

Offset: 0

Greg Dresden and Tianle Tina Yao, Jun 12 2022

nonn

This is the third column in A229556.

			Here is one of the 73 tilings for the n=4 case. Note that vertical dominoes are only allowed in the "first" column.
.  _
  | |_
  |_|_|_
  | |___|_
  |_|_|___|

Cf. A229556, A000045, A003266.

T[0] = 1; T[1] = 1; T[n_] := T[n] = Fibonacci[n + 1] T[n - 1] + (Fibonacci[n] Fibonacci[n - 1]) T[n - 2]; Table[T[n], {n, 0, 20}]

a(n) = Fibonacci(n+1)*a(n-1) + Fibonacci(n)*Fibonacci(n-1)*a(n-2).

a(n) = P(n+1) + Sum_{i=2..n} a(i-1)*Fibonacci(i-1)*Fibonacci(n-(i-1))*P(n)/P(i), where P(n) = A003266(n) the product of the first n Fibonacci numbers. - Greg Dresden and Tianle Tina Yao, Jul 03 2022

cp's OEIS Frontend

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

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