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A374398
Irregular triangle read by rows: j-th exponent in the polynomial P(n,x) in A374385, (exponents in increasing order).
1
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.
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
KEYWORD
nonn,tabl
AUTHOR
Mats Granvik, Jul 07 2024
STATUS
approved