A341835 -id:A341835 - cp's OEIS frontend

A341831 Dirichlet g.f.: 1 / zeta(s)^5.

1, -5, -5, 10, -5, 25, -5, -10, 10, 25, -5, -50, -5, 25, 25, 5, -5, -50, -5, -50, 25, 25, -5, 50, 10, 25, -10, -50, -5, -125, -5, -1, 25, 25, 25, 100, -5, 25, 25, 50, -5, -125, -5, -50, -50, 25, -5, -25, 10, -50, 25, -50, -5, 50, 25, 50, 25, 25, -5, 250, -5, 25, -50, 0, 25

Offset: 1

Ilya Gutkovskiy, Feb 21 2021

Dirichlet inverse of A061200.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Row 5 of A346148.

Cf. A007427, A007428, A008683, A061200, A247343, A341832, A341833, A341834, A341835, A341836.

a[1] = 1; a[n_] := Times @@ ((-1)^#[[2]] Binomial[5, #[[2]]] &/@ FactorInteger[n]); Table[a[n], {n, 65}]

for(n=1, 100, print1(direuler(p=2, n, (1 - X)^5)[n], ", ")) \\ Vaclav Kotesovec, Feb 22 2021

Multiplicative with a(p^e) = (-1)^e * binomial(5, e).

a(1) = 1; a(n) = -Sum_{d|n, d < n} tau_5(n/d) * a(d).

A341832 Dirichlet g.f.: 1 / zeta(s)^6.

Original entry on oeis.org

1, -6, -6, 15, -6, 36, -6, -20, 15, 36, -6, -90, -6, 36, 36, 15, -6, -90, -6, -90, 36, 36, -6, 120, 15, 36, -20, -90, -6, -216, -6, -6, 36, 36, 36, 225, -6, 36, 36, 120, -6, -216, -6, -90, -90, 36, -6, -90, 15, -90, 36, -90, -6, 120, 36, 120, 36, 36, -6, 540, -6, 36, -90, 1, 36

Offset: 1

Ilya Gutkovskiy, Feb 21 2021

Dirichlet inverse of A034695.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Row 6 of A346148.

Cf. A007427, A007428, A008683, A034695, A247343, A341831, A341833, A341834, A341835, A341836.

a[1] = 1; a[n_] := Times @@ ((-1)^#[[2]] Binomial[6, #[[2]]] &/@ FactorInteger[n]); Table[a[n], {n, 65}]

for(n=1, 100, print1(direuler(p=2, n, (1 - X)^6)[n], ", ")) \\ Vaclav Kotesovec, Feb 22 2021

Multiplicative with a(p^e) = (-1)^e * binomial(6, e).

a(1) = 1; a(n) = -Sum_{d|n, d < n} tau_6(n/d) * a(d).

A341833 Dirichlet g.f.: 1 / zeta(s)^7.

Original entry on oeis.org

1, -7, -7, 21, -7, 49, -7, -35, 21, 49, -7, -147, -7, 49, 49, 35, -7, -147, -7, -147, 49, 49, -7, 245, 21, 49, -35, -147, -7, -343, -7, -21, 49, 49, 49, 441, -7, 49, 49, 245, -7, -343, -7, -147, -147, 49, -7, -245, 21, -147, 49, -147, -7, 245, 49, 245, 49, 49, -7, 1029, -7, 49

Offset: 1

Ilya Gutkovskiy, Feb 21 2021

Dirichlet inverse of A111217.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Row 7 of A346148.

Cf. A007427, A007428, A008683, A111217, A247343, A341831, A341832, A341834, A341835, A341836.

a[1] = 1; a[n_] := Times @@ ((-1)^#[[2]] Binomial[7, #[[2]]] &/@ FactorInteger[n]); Table[a[n], {n, 62}]

for(n=1, 100, print1(direuler(p=2, n, (1 - X)^7)[n], ", ")) \\ Vaclav Kotesovec, Feb 22 2021

Multiplicative with a(p^e) = (-1)^e * binomial(7, e).

a(1) = 1; a(n) = -Sum_{d|n, d < n} tau_7(n/d) * a(d).

A341834 Dirichlet g.f.: 1 / zeta(s)^8.

Original entry on oeis.org

1, -8, -8, 28, -8, 64, -8, -56, 28, 64, -8, -224, -8, 64, 64, 70, -8, -224, -8, -224, 64, 64, -8, 448, 28, 64, -56, -224, -8, -512, -8, -56, 64, 64, 64, 784, -8, 64, 64, 448, -8, -512, -8, -224, -224, 64, -8, -560, 28, -224, 64, -224, -8, 448, 64, 448, 64, 64, -8, 1792

Offset: 1

Ilya Gutkovskiy, Feb 21 2021

Dirichlet inverse of A111218.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Row 8 of A346148.

Cf. A007427, A007428, A008683, A111218, A247343, A341831, A341832, A341833, A341835, A341836.

a[1] = 1; a[n_] := Times @@ ((-1)^#[[2]] Binomial[8, #[[2]]] &/@ FactorInteger[n]); Table[a[n], {n, 60}]

for(n=1, 100, print1(direuler(p=2, n, (1 - X)^8)[n], ", ")) \\ Vaclav Kotesovec, Feb 22 2021

Multiplicative with a(p^e) = (-1)^e * binomial(8, e).

a(1) = 1; a(n) = -Sum_{d|n, d < n} tau_8(n/d) * a(d).

A341836 Dirichlet g.f.: 1 / zeta(s)^10.

Original entry on oeis.org

1, -10, -10, 45, -10, 100, -10, -120, 45, 100, -10, -450, -10, 100, 100, 210, -10, -450, -10, -450, 100, 100, -10, 1200, 45, 100, -120, -450, -10, -1000, -10, -252, 100, 100, 100, 2025, -10, 100, 100, 1200, -10, -1000, -10, -450, -450, 100, -10, -2100, 45, -450, 100, -450

Offset: 1

Ilya Gutkovskiy, Feb 21 2021

Dirichlet inverse of A111220.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Row 10 of A346148.

Cf. A007427, A007428, A008683, A111220, A247343, A341831, A341832, A341833, A341834, A341835.

a[1] = 1; a[n_] := Times @@ ((-1)^#[[2]] Binomial[10, #[[2]]] &/@ FactorInteger[n]); Table[a[n], {n, 52}]

for(n=1, 100, print1(direuler(p=2, n, (1 - X)^10)[n], ", ")) \\ Vaclav Kotesovec, Feb 22 2021

Multiplicative with a(p^e) = (-1)^e * binomial(10, e).

a(1) = 1; a(n) = -Sum_{d|n, d < n} tau_10(n/d) * a(d).

A346148 Square array read by descending antidiagonals: T(n, k) = mu^n(k) where mu^1(k) = mu(k) = A008683(k) and for each n >= 1, mu^(n+1)(k) is the Dirichlet convolution of mu(k) and mu^n(k).

Original entry on oeis.org

1, -1, 1, -1, -2, 1, 0, -2, -3, 1, -1, 1, -3, -4, 1, 1, -2, 3, -4, -5, 1, -1, 4, -3, 6, -5, -6, 1, 0, -2, 9, -4, 10, -6, -7, 1, 0, 0, -3, 16, -5, 15, -7, -8, 1, 1, 1, -1, -4, 25, -6, 21, -8, -9, 1, -1, 4, 3, -4, -5, 36, -7, 28, -9, -10, 1, 0, -2, 9, 6, -10, -6

Offset: 1

Sebastian Karlsson, Aug 20 2021

			  n\k| 1    2    3    4    5    6    7    8    9   10   11    12 ...
  ---+--------------------------------------------------------------
   1 | 1   -1   -1    0   -1    1   -1    0    0    1   -1     0 ...
   2 | 1   -2   -2    1   -2    4   -2    0    1    4   -2    -2 ...
   3 | 1   -3   -3    3   -3    9   -3   -1    3    9   -3    -9 ...
   4 | 1   -4   -4    6   -4   16   -4   -4    6   16   -4   -24 ...
   5 | 1   -5   -5   10   -5   25   -5  -10   10   25   -5   -50 ...
   6 | 1   -6   -6   15   -6   36   -6  -20   15   36   -6   -90 ...
   7 | 1   -7   -7   21   -7   49   -7  -35   21   49   -7  -147 ...
   8 | 1   -8   -8   28   -8   64   -8  -56   28   64   -8  -224 ...
   9 | 1   -9   -9   36   -9   81   -9  -84   36   81   -9  -324 ...
  10 | 1  -10  -10   45  -10  100  -10 -120   45  100  -10  -450 ...
  11 | 1  -11  -11   55  -11  121  -11 -165   55  121  -11  -605 ...
  12 | 1  -12  -12   66  -12  144  -12 -220   66  144  -12  -792 ...
  13 | 1  -13  -13   78  -13  169  -13 -286   78  169  -13 -1014 ...
  14 | 1  -14  -14   91  -14  196  -14 -364   91  196  -14 -1274 ...
  15 | 1  -15  -15  105  -15  225  -15 -455  105  225  -15 -1575 ...
  ...

Sebastian Karlsson, Antidiagonals n = 1..140, flattened

Row n=1..10 give A008683, A007427, A007428, A247343, A341831, A341832, A341833, A341834, A341835, A341836.
Main diagonal gives A341837.
Cf. A077592, A163767,

T[n_, k_] := If[k == 1, 1, Product[(-1)^e Binomial[n, e], {e, FactorInteger[k][[All, 2]]}]];
Table[T[n-k+1, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Sep 13 2021 *)

T(n, k) = my(f=factor(k)); for (k=1, #f~, f[k,1] = binomial(n, f[k,2])*(-1)^f[k,2]; f[k,2]=1); factorback(f); \\ Michel Marcus, Aug 21 2021

from sympy import binomial, primefactors as pf, multiplicity as mult
from math import prod
def T(n, k):
    return prod((-1)**mult(p, k)*binomial(n, mult(p, k)) for p in pf(k))

If k = Product (p_j^m_j) then T(n, k) = Product (binomial(n, m_j)*(-1)^m_j).
Dirichlet g.f. of the n-th row: 1/zeta^n(s).
T(n, p) = -n.
T(n, n) = A341837(n).

A341837 If n = Product (p_j^k_j) then a(n) = Product ((-1)^k_j * binomial(n, k_j)).

Original entry on oeis.org

1, -2, -3, 6, -5, 36, -7, -56, 36, 100, -11, -792, -13, 196, 225, 1820, -17, -2754, -19, -3800, 441, 484, -23, 48576, 300, 676, -2925, -10584, -29, -27000, -31, -201376, 1089, 1156, 1225, 396900, -37, 1444, 1521, 395200, -41, -74088, -43, -41624, -44550

Offset: 1

Ilya Gutkovskiy, Feb 21 2021

sign

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A007427, A007428, A008683, A163767, A247343, A341831, A341832, A341833, A341834, A341835, A341836.

Cf. A346148.

a[1] = 1; a[n_] := Times @@ ((-1)^#[[2]] Binomial[n, #[[2]]] &/@ FactorInteger[n]); Table[a[n], {n, 45}]

a(n) = my(f=factor(n)[,2]); prod(k=1, #f, (-1)^f[k]*binomial(n, f[k])); \\ Michel Marcus, Feb 21 2021

a(n) = A346148(n, n). - Sebastian Karlsson, Aug 22 2021

cp's OEIS Frontend

A341831 Dirichlet g.f.: 1 / zeta(s)^5.

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A341832 Dirichlet g.f.: 1 / zeta(s)^6.

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A341833 Dirichlet g.f.: 1 / zeta(s)^7.

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A341834 Dirichlet g.f.: 1 / zeta(s)^8.

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A341836 Dirichlet g.f.: 1 / zeta(s)^10.

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A346148 Square array read by descending antidiagonals: T(n, k) = mu^n(k) where mu^1(k) = mu(k) = A008683(k) and for each n >= 1, mu^(n+1)(k) is the Dirichlet convolution of mu(k) and mu^n(k).

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A341837 If n = Product (p_j^k_j) then a(n) = Product ((-1)^k_j * binomial(n, k_j)).

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