A345229 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals, where T(n,k) = Sum_{1 <= x_1 <= x_2 <= ... <= x_k <= n} gcd(x_1, x_2, ..., x_k).
1, 1, 3, 1, 4, 6, 1, 5, 9, 10, 1, 6, 13, 17, 15, 1, 7, 18, 28, 26, 21, 1, 8, 24, 44, 47, 41, 28, 1, 9, 31, 66, 83, 82, 54, 36, 1, 10, 39, 95, 140, 159, 116, 74, 45, 1, 11, 48, 132, 225, 293, 249, 172, 95, 55, 1, 12, 58, 178, 346, 512, 509, 401, 235, 122, 66, 1, 13, 69, 234, 512, 852, 980, 888, 592, 321, 143, 78
Offset: 1
Examples
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, ...
Links
- Seiichi Manyama, Antidiagonals n = 1..140, flattened
Crossrefs
Programs
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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 *) -
PARI
T(n, k) = sum(j=1, n, sumdiv(j, d, eulerphi(j/d)*binomial(d+k-2, k-1)));
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PARI
T(n, k) = sum(j=1, n, eulerphi(j)*binomial(n\j+k-1, k)); \\ Seiichi Manyama, Sep 13 2024
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