cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A038063 Product_{k>=1}1/(1 - x^k)^a(k) = 1 + 2x.

Original entry on oeis.org

2, -3, 2, -3, 6, -11, 18, -30, 56, -105, 186, -335, 630, -1179, 2182, -4080, 7710, -14588, 27594, -52377, 99858, -190743, 364722, -698870, 1342176, -2581425, 4971008, -9586395, 18512790, -35792449, 69273666, -134215680, 260300986
Offset: 1

Views

Author

Christian G. Bower, Jan 04 1999

Keywords

Comments

Apart from initial terms, exponents in expansion of A065472 as a product zeta(n)^(-a(n)).

Links

Crossrefs

Cf. A001037, A008683, A038064, A038065, A038066, A038067, A038068, A038069, A038070, A059966, A060477, A065472.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, (-1)^(#+1) * MoebiusMu[n/#]*2^# &] / n; Array[a, 33] (* Amiram Eldar, May 29 2025 *)
  • PARI
    {a(n)=polcoeff(sum(k=1,n,moebius(k)/k*log(1+2*x^k+x*O(x^n))),n)} \\ Paul D. Hanna, Oct 13 2010

Formula

a(n) = (1/n) * Sum_{d divides n} (-1)^(d+1)*moebius(n/d)*2^d. - Vladeta Jovovic, Sep 06 2002
G.f.: Sum_{n>=1} moebius(n)*log(1 + 2*x^n)/n, where moebius(n) = A008683(n). - Paul D. Hanna, Oct 13 2010
For n == 0, 1, 3 (mod 4), a(n) = (-1)^(n+1)*A001037(n), which for n>1 also equals (-1)^(n+1)*A059966(n) = (-1)^(n+1)*A060477(n).
For n == 2 (mod 4), a(n) = -(A001037(n) + A001037(n/2)). - George Beck and Max Alekseyev, May 23 2016
a(n) ~ -(-1)^n * 2^n / n. - Vaclav Kotesovec, Jun 12 2018

A038070 Product_{k>=1} (1+x^k)^a(k) = 1 + 5x.

Original entry on oeis.org

5, -10, 40, -160, 624, -2580, 11160, -48910, 217000, -976248, 4438920, -20346280, 93900240, -435959820, 2034504992, -9536767660, 44878791360, -211927516500, 1003867701480, -4768372069968, 22706531339280, -108372079190340, 518301258916440, -2483526875798780
Offset: 1

Views

Author

Christian G. Bower, Jan 04 1999

Keywords

Links

Crossrefs

Cf. A038063, A038064, A038065, A038066, A038067, A038068, A038069, A306159.

Formula

Dirichlet convolution of A038066 with characteristic function of powers of 2.
a(n) = (1/n) * (-(-5)^n + Sum_{dSeiichi Manyama, Jun 23 2018

A038064 Product_{k>=1} 1/(1 - x^k)^a(k) = 1 + 3x.

Original entry on oeis.org

3, -6, 8, -18, 48, -124, 312, -810, 2184, -5928, 16104, -44220, 122640, -341796, 956576, -2690010, 7596480, -21524412, 61171656, -174336264, 498111952, -1426419852, 4093181688, -11767874940, 33891544368, -97764131640
Offset: 1

Views

Author

Christian G. Bower, Jan 04 1999

Keywords

Links

Crossrefs

Cf. A038063, A038064, A038065, A038066, A038067, A038068, A038069, A038070.

Programs

  • PARI
    {a(n)=polcoeff(sum(k=1,n,moebius(k)/k*log(1+3*x^k+x*O(x^n))),n)} \\ Paul D. Hanna, Oct 13 2010

Formula

G.f.: Sum_{n>=1} moebius(n)*log(1 + 3*x^n)/n, where moebius(n)=A008683(n). - Paul D. Hanna, Oct 13 2010
a(n) = -(1/n) * Sum_{d|n} mu(n/d) * (-3)^d. - Seiichi Manyama, Apr 12 2025

A038065 Product_{k>=1} 1/(1 - x^k)^a(k) = 1 + 4x.

Original entry on oeis.org

4, -10, 20, -60, 204, -690, 2340, -8160, 29120, -104958, 381300, -1397740, 5162220, -19175130, 71582716, -268431360, 1010580540, -3817763040, 14467258260, -54975528948, 209430785460, -799645010850, 3059510616420
Offset: 1

Views

Author

Christian G. Bower, Jan 04 1999

Keywords

Links

Crossrefs

Cf. A038063, A038064, A038066, A038067, A038068, A038069, A038070.

Programs

  • PARI
    x='x+O('x^24); Vec(sum(k=1, 24, moebius(k)*log(1 + 4*x^k)/k)) \\ Indranil Ghosh, May 24 2017

Formula

G.f.: Sum_{k>=1} mu(k)*log(1 + 4*x^k)/k. - Ilya Gutkovskiy, May 23 2017
a(n) ~ -(-1)^n * 4^n / n. - Vaclav Kotesovec, Jun 12 2018
a(n) = -(1/n) * Sum_{d|n} mu(n/d) * (-4)^d. - Seiichi Manyama, Apr 12 2025

A343467 a(n) = -(1/n) * Sum_{d|n} phi(n/d) * (-5)^d.

Original entry on oeis.org

5, -10, 45, -160, 629, -2590, 11165, -48910, 217045, -976258, 4438925, -20346440, 93900245, -435959830, 2034505661, -9536767660, 44878791365, -211927519090, 1003867701485, -4768372070128, 22706531350485, -108372079190350, 518301258916445, -2483526875847690, 11920928955078629
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 16 2021

Keywords

Crossrefs

Cf. A000010, A001869, A038066, A074763, A261569, A343465, A343466.

Programs

  • Mathematica
    Table[-(1/n) Sum[EulerPhi[n/d] (-5)^d, {d, Divisors[n]}], {n, 1, 25}]
nmax = 25; CoefficientList[Series[Sum[EulerPhi[k] Log[1 + 5 x^k]/k, {k, 1, nmax}], {x, 0, nmax}], x] // Rest

Formula

G.f.: Sum_{k>=1} phi(k) * log(1 + 5*x^k) / k.
a(n) = -(1/n) * Sum_{k=1..n} (-5)^gcd(n,k).
Product_{n>=1} 1 / (1 - x^n)^a(n) = g.f. for A261569.

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

Views

Author

Seiichi Manyama, Apr 12 2025

Keywords

Examples

			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, ...

Crossrefs

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

Programs

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

Formula

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.
Showing 1-6 of 6 results.

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