cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

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).

Original entry on oeis.org

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

Views

Author

Seiichi Manyama, Apr 12 2025

Keywords

Examples

			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, ...

Crossrefs

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

Programs

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

Formula

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).

A383003 a(n) = Sum_{d|n} (-n)^(d-1).

Original entry on oeis.org

1, -1, 10, -67, 626, -7745, 117650, -2097671, 43046803, -999990009, 25937424602, -743008621115, 23298085122482, -793714765724621, 29192926025441476, -1152921504875286543, 48661191875666868482, -2185911559727678460653, 104127350297911241532842
Offset: 1

Views

Author

Seiichi Manyama, Apr 12 2025

Keywords

Crossrefs

Cf. A101561, A101562, A101563.
Cf. A262843, A308814, A383010.
Cf. A382998, A383012.

Programs

  • PARI
    a(n) = sumdiv(n, d, (-n)^(d-1));

Formula

a(n) = (1/n) * A383010(n).
a(n) = [x^n] Sum_{k>=1} log(1 + n*x^k) / k.
a(n) = [x^n] Sum_{k>=1} x^k / (1 + n*x^k).

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

Views

Author

Seiichi Manyama, Apr 11 2025

Keywords

Examples

			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, ...

Crossrefs

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

Programs

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

Formula

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).

A383012 a(n) = Sum_{d|n} mu(n/d) * (-n)^(d-1).

Original entry on oeis.org

1, -3, 8, -60, 624, -7805, 117648, -2096640, 43046640, -1000009989, 25937424600, -743008120140, 23298085122480, -793714780783665, 29192926025339776, -1152921504338411520, 48661191875666868480, -2185911559749714602652, 104127350297911241532840
Offset: 1

Views

Author

Seiichi Manyama, Apr 12 2025

Keywords

Crossrefs

Main diagonal of A383011.
Cf. A008683, A075147, A382998, A383003.

Programs

  • PARI
    a(n) = sumdiv(n, d, moebius(n/d)*(-n)^(d-1));

Formula

a(n) = [x^n] Sum_{k>=1} mu(k) * log(1 + n*x^k) / k.
a(n) = [x^n] Sum_{k>=1} mu(k) * x^k / (1 + n*x^k).
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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