A345036 -id:A345036 - cp's OEIS frontend

A345033 Square array T(n,k), n >= 1, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=1..n} (-k)^(floor(n/j) - 1).

1, 1, 1, 1, 0, 2, 1, -1, 3, 2, 1, -2, 6, 0, 3, 1, -3, 11, -8, 3, 3, 1, -4, 18, -28, 17, 2, 4, 1, -5, 27, -66, 81, -27, 5, 4, 1, -6, 38, -128, 255, -234, 70, 0, 5, 1, -7, 51, -220, 623, -1009, 739, -136, 5, 5, 1, -8, 66, -348, 1293, -3102, 4112, -2216, 255, 4, 6, 1, -9, 83, -518, 2397, -7743, 15649, -16452, 6545, -491, 7, 6

Offset: 1

Seiichi Manyama, Jun 06 2021

			Square array begins:
  1, 1,   1,    1,     1,     1,     1, ...
  1, 0,  -1,   -2,    -3,    -4,    -5, ...
  2, 3,   6,   11,    18,    27,    38, ...
  2, 0,  -8,  -28,   -66,  -128,  -220, ...
  3, 3,  17,   81,   255,   623,  1293, ...
  3, 2, -27, -234, -1009, -3102, -7743, ...

Columns k=0..3 give A110654, A271860, A345034, A345035.
T(n,n) gives A345036.
Cf. A344824, A345032.

T[n_, 0] := Floor[(n + 1)/2]; T[n_, k_] := Sum[(-k)^(Floor[n/j] - 1), {j, 1, n}]; Table[T[k, n - k], {n, 1, 12}, {k, 1, n}] // Flatten (* Amiram Eldar, Jun 06 2021 *)

PARI
```
T(n, k) = sum(j=1, n, (-k)^(n\j-1));
```

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

A345030 a(n) = Sum_{k=1..n} n^(floor(n/k) - 1).

Original entry on oeis.org

1, 3, 11, 70, 633, 7821, 117709, 2097684, 43047545, 1000010125, 25937439391, 743008621422, 23298085496173, 793714780786669, 29192926036832363, 1152921504875352376, 48661191876077295937, 2185911559749718388655, 104127350297928227579629

Offset: 1

Seiichi Manyama, Jun 06 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..387

Diagonal of A345032.

Cf. A023037, A332533, A344551, A345036.

a[n_] := Sum[n^(Floor[n/k] - 1), {k, 1, n}]; Array[a, 20] (* Amiram Eldar, Jun 06 2021 *)

PARI
```
a(n) = sum(k=1, n, n^(n\k-1));
```

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

A345037 a(n) = Sum_{k=1..n} (-k)^(floor(n/k) - 1).

Original entry on oeis.org

1, 0, 3, -1, 2, 3, 6, -12, 3, 20, 23, -49, -46, 41, 182, -100, -97, -6, -3, -613, 418, 1941, 1944, -5518, -4765, 1364, 10205, 2629, 2632, -1181, -1178, -71404, 7463, 105748, 127245, -233385, -233382, 159813, 868586, -335790, -335787, -853276, -853273, -2689757, 4163818

Offset: 1

Seiichi Manyama, Jun 06 2021

sign

Seiichi Manyama, Table of n, a(n) for n = 1..5000

Cf. A344551, A345036.

a[n_] := Sum[(-k)^(Floor[n/k] - 1), {k, 1, n}]; Array[a, 45] (* Amiram Eldar, Jun 06 2021 *)

PARI
```
a(n) = sum(k=1, n, (-k)^(n\k-1));
```

my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, x^k*(1-x^k)/(1+k*x^k))/(1-x))

G.f.: (1/(1 - x)) * Sum_{k>=1} x^k * (1 - x^k)/(1 + k*x^k).

|a(n)| ~ 3^((n - mod(n,3))/3 - 1). - Vaclav Kotesovec, Jun 12 2021

cp's OEIS Frontend

A345033 Square array T(n,k), n >= 1, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=1..n} (-k)^(floor(n/j) - 1).

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A345030 a(n) = Sum_{k=1..n} n^(floor(n/k) - 1).

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A345037 a(n) = Sum_{k=1..n} (-k)^(floor(n/k) - 1).

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