OFFSET
1,3
LINKS
Seiichi Manyama, Antidiagonals n = 1..140, flattened
FORMULA
G.f. of column k: (1/(1 - x)) * Sum_{j>=1} phi(j) * x^j/(1 - x^j)^k.
T(n,k) = Sum_{j=1..n} Sum_{d|j} phi(j/d) * binomial(d+k-2, k-1).
T(n,k) = Sum_{j=1..n} phi(j) * binomial(floor(n/j)+k-1,k). - Seiichi Manyama, Sep 13 2024
EXAMPLE
G.f. of column 3: (1/(1 - x)) * Sum_{j>=1} phi(j) * x^j/(1 - x^j)^3.
Square array begins:
1, 1, 1, 1, 1, 1, 1, ...
3, 4, 5, 6, 7, 8, 9, ...
6, 9, 13, 18, 24, 31, 39, ...
10, 17, 28, 44, 66, 95, 132, ...
15, 26, 47, 83, 140, 225, 346, ...
21, 41, 82, 159, 293, 512, 852, ...
28, 54, 116, 249, 509, 980, 1782, ...
MAPLE
T:= (n, k)-> coeff(series((1/(1-x))* add(numtheory[phi](j)
*x^j/(1-x^j)^k, j=1..n), x, n+1), x, n):
seq(seq(T(n, 1+d-n), n=1..d), d=1..12); # Alois P. Heinz, Jun 11 2021
MATHEMATICA
T[n_, k_] := Sum[DivisorSum[j, EulerPhi[j/#] * Binomial[k + # - 2, k - 1] &], {j, 1, n}]; Table[T[k, n - k + 1], {n, 1, 12}, {k, 1, n}] // Flatten (* Amiram Eldar, Jun 11 2021 *)
PROG
(PARI) T(n, k) = sum(j=1, n, sumdiv(j, d, eulerphi(j/d)*binomial(d+k-2, k-1)));
(PARI) T(n, k) = sum(j=1, n, eulerphi(j)*binomial(n\j+k-1, k)); \\ Seiichi Manyama, Sep 13 2024
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Seiichi Manyama, Jun 11 2021
STATUS
approved