A383012 -id:A383012 - cp's OEIS frontend

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

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

Offset: 1

Seiichi Manyama, Apr 12 2025

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

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

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).

A383011 Square array A(n,k), n >= 1, k >= 1, read by antidiagonals downwards, where A(n,k) = -(1/n) * Sum_{d|n} mu(n/d) * (-k)^d.

Original entry on oeis.org

1, 2, -1, 3, -3, 0, 4, -6, 2, 0, 5, -10, 8, -3, 0, 6, -15, 20, -18, 6, 0, 7, -21, 40, -60, 48, -11, 0, 8, -28, 70, -150, 204, -124, 18, 0, 9, -36, 112, -315, 624, -690, 312, -30, 0, 10, -45, 168, -588, 1554, -2620, 2340, -810, 56, 0, 11, -55, 240, -1008, 3360, -7805, 11160, -8160, 2184, -105, 0

Offset: 1

Seiichi Manyama, Apr 12 2025

			Square array begins:
   1,   2,    3,    4,     5,     6,      7, ...
  -1,  -3,   -6,  -10,   -15,   -21,    -28, ...
   0,   2,    8,   20,    40,    70,    112, ...
   0,  -3,  -18,  -60,  -150,  -315,   -588, ...
   0,   6,   48,  204,   624,  1554,   3360, ...
   0, -11, -124, -690, -2620, -7805, -19656, ...
   0,  18,  312, 2340, 11160, 39990, 117648, ...

Columns k=1..5 give A154955, A038063, A038064, A038065, A038066.
Main diagonal gives A383012.
Cf. A008683, A074650.

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

G.f. of column k: Sum_{j>=1} mu(j) * log(1 + k*x^j) / j.

Product_{n>=1} 1/(1 - x^n)^A(n,k) = 1 + k*x.

cp's OEIS Frontend

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

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