A101562 -id:A101562 - cp's OEIS frontend

A344819 a(n) = Sum_{k=1..n} floor(n/k) * (-4)^(k-1).

1, -2, 15, -52, 205, -806, 3291, -13160, 52393, -209498, 839079, -3356300, 13420917, -53683854, 214751875, -859006400, 3435960897, -13743843762, 54975632975, -219902535924, 879609095965, -3518436366566, 14073749677851, -56294998711576, 225179977999337, -900719912066074

Offset: 1

Seiichi Manyama, May 29 2021

sign

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Column k=4 of A344824.

Cf. A059851, A101562, A268235, A332533, A344814, A344815, A344816, A344817, A344818, A344820.

A344819:= func< n | (&+[(-4)^(k-1)*Floor(n/k): k in [1..n]]) >;
[A344819(n): n in [1..40]]; // G. C. Greubel, Jun 25 2024

a[n_] := Sum[(-4)^(k - 1) * Quotient[n, k], {k, 1, n}]; Array[a, 30] (* Amiram Eldar, May 29 2021 *)

PARI
```
a(n) = sum(k=1, n, n\k*(-4)^(k-1));
```

a(n) = sum(k=1, n, sumdiv(k, d, (-4)^(d-1)));

my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, x^k/(1+4*x^k))/(1-x))

my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, (-4)^(k-1)*x^k/(1-x^k))/(1-x))

def A344819(n): return sum((-4)^(k-1)*int(n//k) for k in range(1,n+1))
[A344819(n) for n in range(1,41)] # G. C. Greubel, Jun 25 2024

a(n) = Sum_{k=1..n} Sum_{d|k} (-4)^(d-1).
G.f.: (1/(1 - x)) * Sum_{k>=1} x^k/(1 + 4*x^k).
G.f.: (1/(1 - x)) * Sum_{k>=1} (-4)^(k-1) * x^k/(1 - x^k).
a(n) ~ -(-1)^n * 4^n / 5. - Vaclav Kotesovec, Jun 05 2021

A101561 a(n) = (-1)^n * [x^n] Sum_{k>=1} x^(k-1)/(1+3*x^k).

Original entry on oeis.org

1, 2, 10, 29, 82, 236, 730, 2216, 6571, 19604, 59050, 177410, 531442, 1593596, 4783060, 14351123, 43046722, 129133838, 387420490, 1162281098, 3486785140, 10460294156, 31381059610, 94143358424, 282429536563, 847288078004, 2541865834900, 7625599078610

Offset: 0

Paul Barry, Dec 07 2004

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A048272, A051731, A081295, A101562, A101563.

A101561:= func< n | (&+[(-1)^(n-k)*3^k*0^((n+1) mod (k+1)): k in [0..n]]) >;
[A101561(n): n in [0..40]]; // G. C. Greubel, Jun 25 2024

a[n_]:= Sum[(-1)^(n-k) * If[Mod[n+1, k+1]==0, 1, 0] * 3^k, {k, 0, n}];
Table[a[n], {n, 0, 25}] (* James C. McMahon, Jan 01 2024 *)
A101561[n_]:= (-1)^n*DivisorSum[n+1, (-3)^(#-1) &];
Table[A101561[n], {n,0,40}] (* G. C. Greubel, Jun 25 2024 *)

def A101561(n): return sum((-1)^(n+k)*3^k*0^((n+1)%(k+1)) for k in range(n+1))
[A101561(n) for n in range(41)] # G. C. Greubel, Jun 25 2024

a(n) = Sum_{k=0..n} (-1)^(n-k) * 3^k * A051731(n+1, k+1).

a(n) = (-1)^n * Sum_{d|n+1} (-3)^(d-1). - G. C. Greubel, Jun 25 2024

A101563 a(n) = (-1)^n * coefficient of x^n in Sum_{k>=1} x^(k-1)/(1+10*x^k).

Original entry on oeis.org

1, 9, 101, 1009, 10001, 99909, 1000001, 10001009, 100000101, 999990009, 10000000001, 100000100909, 1000000000001, 9999999000009, 100000000010101, 1000000010001009, 10000000000000001, 99999999900099909

Offset: 0

Paul Barry, Dec 07 2004

G. C. Greubel, Table of n, a(n) for n = 0..990

Cf. A051731, A081295, A048272, A101561, A101562.

A101563:= func< n | (&+[(-1)^(n-k)*(10)^k*0^((n+1) mod (k+1)): k in [0..n]]) >;
[A101563(n): n in [0..40]]; // G. C. Greubel, Jun 25 2024

A101563[n_]:= (-1)^n*DivisorSum[n+1, (-10)^(#-1) &];
Table[A101563[n], {n,0,40}] (* G. C. Greubel, Jun 25 2024 *)

def A101563(n): return sum((-1)^(n+k)*(10)^k*0^((n+1)%(k+1)) for k in range(n+1))
[A101563(n) for n in range(41)] # G. C. Greubel, Jun 25 2024

a(n) = Sum_{k=0..n} (-1)^(n-k) * (10)^k * A051731(n+1, k+1).

a(n) = (-1)^n * Sum_{d|n+1} (-10)^(d-1). - G. C. Greubel, Jun 25 2024

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

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).

cp's OEIS Frontend

A344819 a(n) = Sum_{k=1..n} floor(n/k) * (-4)^(k-1).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

PARI

PARI

SageMath

Formula

A101561 a(n) = (-1)^n * [x^n] Sum_{k>=1} x^(k-1)/(1+3*x^k).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

A101563 a(n) = (-1)^n * coefficient of x^n in Sum_{k>=1} x^(k-1)/(1+10*x^k).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

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

Views

Author

Keywords

Crossrefs

Programs

PARI

Formula