A382994 -id:A382994 - cp's OEIS frontend

A382995 Square array A(n,k), n >= 1, k >= 1, read by antidiagonals downwards, where A(n,k) = Sum_{d|n} phi(n/d) * (-k)^(d-1).

1, 1, 0, 1, -1, 3, 1, -2, 6, 0, 1, -3, 11, -8, 5, 1, -4, 18, -28, 20, 0, 1, -5, 27, -66, 85, -30, 7, 1, -6, 38, -128, 260, -238, 70, 0, 1, -7, 51, -220, 629, -1014, 735, -136, 9, 1, -8, 66, -348, 1300, -3108, 4102, -2216, 270, 0, 1, -9, 83, -518, 2405, -7750, 15631, -16452, 6585, -500, 11

Offset: 1

Seiichi Manyama, Apr 12 2025

			Square array begins:
  1,   1,    1,     1,     1,     1,      1, ...
  0,  -1,   -2,    -3,    -4,    -5,     -6, ...
  3,   6,   11,    18,    27,    38,     51, ...
  0,  -8,  -28,   -66,  -128,  -220,   -348, ...
  5,  20,   85,   260,   629,  1300,   2405, ...
  0, -30, -238, -1014, -3108, -7750, -16770, ...
  7,  70,  735,  4102, 15631, 46662, 117655, ...

Columns k=1..3 give A193356, A382999, A383000.
Main diagonal gives A382998.
Cf. A000010, A343489, A382994.

a(n, k) = sumdiv(n, d, eulerphi(n/d)*(-k)^(d-1));

A(n,k) = (1/k) * A382994(n,k).
A(n,k) = Sum_{j=1..n} (-k)^(gcd(n,j) - 1).
G.f. of column k: Sum_{j>=1} phi(j) * x^j / (1 + k*x^j).

A382993 Square array A(n,k), n >= 1, k >= 1, read by antidiagonals downwards, where A(n,k) = -(1/n) * Sum_{d|n} phi(n/d) * (-k)^d.

Original entry on oeis.org

1, 2, 0, 3, -1, 1, 4, -3, 4, 0, 5, -6, 11, -4, 1, 6, -10, 24, -21, 8, 0, 7, -15, 45, -66, 51, -10, 1, 8, -21, 76, -160, 208, -119, 20, 0, 9, -28, 119, -330, 629, -676, 315, -34, 1, 10, -36, 176, -609, 1560, -2590, 2344, -831, 60, 0, 11, -45, 249, -1036, 3367, -7750, 11165, -8226, 2195, -100, 1

Offset: 1

Seiichi Manyama, Apr 11 2025

			Square array begins:
  1,   2,    3,    4,     5,     6,      7, ...
  0,  -1,   -3,   -6,   -10,   -15,    -21, ...
  1,   4,   11,   24,    45,    76,    119, ...
  0,  -4,  -21,  -66,  -160,  -330,   -609, ...
  1,   8,   51,  208,   629,  1560,   3367, ...
  0, -10, -119, -676, -2590, -7750, -19565, ...
  1,  20,  315, 2344, 11165, 39996, 117655, ...

Columns k=1..5 give A000035, (-1)^(n+1) * A074763(n), A343465, A343466, A343467.
Main diagonal gives A382998.
Cf. A000010, A075195, A286957, A382994, A383011.

a(n, k) = -sumdiv(n, d, eulerphi(n/d)*(-k)^d)/n;

A(n,k) = (1/n) * A382994(n,k).
A(n,k) = -(1/n) * Sum_{j=1..n} (-k)^gcd(n,j).
G.f. of column k: Sum_{j>=1} phi(j) * log(1 + k*x^j) / j.
Product_{n>=1} 1/(1 - x^n)^A(n,k) = Product_{n>=1} (1 + k*x^n).

A382997 a(n) = -Sum_{d|n} phi(n/d) * (-n)^d.

Original entry on oeis.org

1, -2, 33, -264, 3145, -46500, 823585, -16781408, 387422001, -9999900360, 285311670721, -8916103472496, 302875106592409, -11112006720145604, 437893890382391745, -18446744078004650880, 827240261886336764449, -39346408075098246299676, 1978419655660313589124321

Offset: 1

Seiichi Manyama, Apr 12 2025

sign

Main diagonal of A382994.

Cf. A000010, A383010.

a(n) = -sumdiv(n, d, eulerphi(n/d)*(-n)^d);

a(n) = -Sum_{k=1..n} (-n)^gcd(n,k).

cp's OEIS Frontend

A382995 Square array A(n,k), n >= 1, k >= 1, read by antidiagonals downwards, where A(n,k) = Sum_{d|n} phi(n/d) * (-k)^(d-1).

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A382993 Square array A(n,k), n >= 1, k >= 1, read by antidiagonals downwards, where A(n,k) = -(1/n) * Sum_{d|n} phi(n/d) * (-k)^d.

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Examples

Crossrefs

Programs

PARI

Formula

A382997 a(n) = -Sum_{d|n} phi(n/d) * (-n)^d.

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Programs

PARI

Formula