A374398 Irregular triangle read by rows: j-th exponent in the polynomial P(n,x) in A374385, (exponents in increasing order).

0, 0, 1, 0, 1, 0, 1, 2, 0, 1, 0, 1, 2, 3, 4, 0, 1, 2, 0, 1, 2, 3, 4, 5, 6, 0, 1, 0, 1, 2, 0, 1, 2, 3, 4, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 0, 1, 2, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0, 1, 2, 3, 4, 5, 6, 0, 1, 3, 4, 5, 7, 8, 0, 1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16

Offset

0,8

Comments

See A374385.

Links

Table of n, a(n) for n=0..94.

Formula

P(n,x) = denominator(Sum_{h=0..oo} Sum_{k=1..n} A023900(GCD(n,k))*x^(n*h + k)).

a(n,m) = j-th exponent of: [x^m] P(n,x), n >= 0, 0 <= m <= abs(A023900(n)).

Example

P(0,x) = x^0; P(1,x) = x^0 - x^1; P(2,x) = x^0 + x^1; P(3,x) = x^0 + x^1 + x^2; P(4,x) = x^0 + x^1; P(5,x) = x^0 + x^1 + x^2 + x^3 + x^4; ...
The irregular triangle a(n,m) begins:
  n\m 0  1  2  3  4  5  6  7  8  9 10 11 12 ...
  0:  0
  1:  0  1
  2:  0  1
  3:  0  1  2
  4:  0  1
  5:  0  1  2  3  4
  6:  0  1  2
  7:  0  1  2  3  4  5  6
  8:  0  1
  9:  0  1  2
 10:  1  2  3  4
 11:  0  1  2  3  4  5  6  7  8  9 10
 12:  0  1  2
 13:  0  1  2  3  4  5  6  7  8  9 10 11 12
 14:  0  1  2  3  4  5  6
 15:  0  1  3  4  5  7  8
 16:  0  1
 ...

Mathematica

nn = 17; f[n_] := DivisorSum[n, MoebiusMu[#]  # &]; Flatten[Table[Exponent[Denominator[Sum[Sum[f[GCD[n, k]]*x^(n*h + k), {k, 1, n}], {h, 0, Infinity}]], x, List], {n, 0, nn}]]

Crossrefs

Cf. A374385, A013596, A013595, A023900, A173557.

Sequence in context: A340499 A336033 A238405 * A004173 A185370 A352747

Adjacent sequences: A374395 A374396 A374397 * A374399 A374400 A374401

Keyword

nonn,tabl

Author

Mats Granvik, Jul 07 2024

Status

approved