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.

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A321438 a(n) = Sum_{d|n} (-1)^(n/d+1)*d^n.

Original entry on oeis.org

1, 3, 28, 239, 3126, 45990, 823544, 16711423, 387440173, 9990235398, 285311670612, 8913939907598, 302875106592254, 11111328602501550, 437893920912786408, 18446462594437808127, 827240261886336764178, 39346258082220810086373, 1978419655660313589123980, 104857499999905732078938574
Offset: 1

Views

Author

Ilya Gutkovskiy, Nov 09 2018

Keywords

Links

Crossrefs

Cf. A000593, A023887, A078306, A078307, A186633, A284900, A284926, A284927, A321385.

Programs

  • Magma
    m:=20; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&+[(k*x)^k/(1+(k*x)^k): k in [1..m]]) ));  // G. C. Greubel, Nov 11 2018
  • Mathematica
    Table[Sum[(-1)^(n/d + 1) d^n, {d, Divisors[n]}], {n, 20}]
nmax = 20; Rest[CoefficientList[Series[Sum[(k x)^k/(1 + (k x)^k), {k, 1, nmax}], {x, 0, nmax}], x]]
nmax = 20; Rest[CoefficientList[Series[Log[Product[(1 + k^k x^k)^(1/k), {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]]
  • PARI
    a(n) = sumdiv(n, d, (-1)^(n/d+1)*d^n); \\ Michel Marcus, Nov 09 2018

Formula

G.f.: Sum_{k>=1} (k*x)^k/(1 + (k*x)^k).
L.g.f.: log(Product_{k>=1} (1 + k^k*x^k)^(1/k)) = Sum_{n>=1} a(n)*x^n/n.
a(n) ~ n^n. - Vaclav Kotesovec, Nov 10 2018

A354508 a(n) = n! * Sum_{k=1..n} ( Sum_{d|k} (-1)^(k/d+1) * d^2 )/(k * (n-k)!).

Original entry on oeis.org

1, 5, 32, 168, 1189, 8785, 77384, 646296, 7306737, 79997893, 1005481784, 12518370128, 184109233125, 2671256865121, 47934480000112, 754158322407248, 13813898274148737, 262680987222463269, 5518034466415262320, 107988236156057411096, 2605128008760639636677
Offset: 1

Views

Author

Seiichi Manyama, Aug 15 2022

Keywords

Crossrefs

Cf. A354506, A354507.
Cf. A078306, A356391.

Programs

  • PARI
    a(n) = n!*sum(k=1, n, sumdiv(k, d, (-1)^(k/d+1)*d^2)/(k*(n-k)!));
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(-exp(x)*sum(k=1, N, (-x)^k/(k*(1-x^k)^2))))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x)*sum(k=1, N, k*log(1+x^k))))

Formula

a(n) = n! * Sum_{k=1..n} A078306(k)/(k * (n-k)!).
E.g.f.: -exp(x) * Sum_{k>0} (-x)^k/(k * (1 - x^k)^2).
E.g.f.: exp(x) * Sum_{k>0} k * log(1 + x^k).

A356566 Expansion of e.g.f. ( Product_{k>0} (1+x^k)^k )^x.

Original entry on oeis.org

1, 0, 2, 9, 92, 510, 7074, 68040, 1002224, 12529944, 228706920, 3565888920, 71035245192, 1348127454960, 30270949077264, 661700017709640, 16516072112482560, 408336559236083520, 11204399270843020224, 309489391954850336640, 9258803420755891835520
Offset: 0

Views

Author

Seiichi Manyama, Aug 12 2022

Keywords

Crossrefs

Cf. A356564, A356565.
Cf. A007113, A078306, A356394.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(prod(k=1, N, (1+x^k)^k)^x))
  • PARI
    a078306(n) = sumdiv(n, d, (-1)^(n/d+1)*d^2);
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=2, i, j!*a078306(j-1)/(j-1)*binomial(i-1, j-1)*v[i-j+1])); v;

Formula

a(0) = 1, a(1) = 0; a(n) = Sum_{k=2..n} k! * A078306(k-1)/(k-1) * binomial(n-1,k-1) * a(n-k).

A326125 Expansion of Sum_{k>=1} k^2 * x^k / (1 + x^k)^2.

Original entry on oeis.org

1, 2, 12, 4, 30, 24, 56, 8, 117, 60, 132, 48, 182, 112, 360, 16, 306, 234, 380, 120, 672, 264, 552, 96, 775, 364, 1080, 224, 870, 720, 992, 32, 1584, 612, 1680, 468, 1406, 760, 2184, 240, 1722, 1344, 1892, 528, 3510, 1104, 2256, 192, 2793, 1550, 3672, 728, 2862, 2160
Offset: 1

Views

Author

Ilya Gutkovskiy, Sep 10 2019

Keywords

Links

Crossrefs

Cf. A000593, A064987, A078306, A143520, A185152, A245579, A326238, A353908.

Programs

  • Mathematica
    nmax = 54; CoefficientList[Series[Sum[k^2 x^k/(1 + x^k)^2, {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[n Sum[(-1)^(n/d + 1) d, {d, Divisors[n]}], {n, 1, 54}]
f[p_, e_] := p^e*(p^(e+1)-1)/(p-1); f[2, e_] := 2^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Dec 05 2022 *)
  • PARI
    a(n)={n*sumdiv(n, d, (-1)^(n/d+1)*d)} \\ Andrew Howroyd, Sep 10 2019

Formula

G.f.: Sum_{k>=1} (-1)^(k + 1) * k * x^k * (1 + x^k) / (1 - x^k)^3.
a(n) = n * Sum_{d|n} (-1)^(n/d + 1) * d.
a(n) = n * A000593(n).
From Amiram Eldar, Dec 05 2022: (Start)
Multiplicative with a(2^e) = 2^e, and a(p^e) = p^e*(p^(e+1)-1)/(p-1) for p > 2.
Sum_{k=1..n} a(k) ~ c * n^3, where c = Pi^2/36 = 0.2741556... (A353908). (End)
Dirichlet g.f.: zeta(s-1)*zeta(s-2)*(1-2^(2-s)). - Amiram Eldar, Jan 07 2023

A372625 Expansion of Sum_{k>=1} k^2 * x^(k^2) / (1 + x^k).

Original entry on oeis.org

1, -1, 1, 3, 1, -5, 1, 3, 10, -5, 1, -6, 1, -5, 10, 19, 1, -14, 1, -13, 10, -5, 1, 10, 26, -5, 10, -13, 1, -39, 1, 19, 10, -5, 26, 14, 1, -5, 10, -6, 1, -50, 1, -13, 35, -5, 1, 46, 50, -30, 10, -13, 1, -50, 26, -30, 10, -5, 1, -11, 1, -5, 59, 83, 26, -50, 1, -13, 10, -79
Offset: 1

Views

Author

Ilya Gutkovskiy, May 07 2024

Keywords

Crossrefs

Cf. A064027, A066839, A078306, A095118, A321543, A321558, A348608.

Programs

  • Mathematica
    nmax = 70; CoefficientList[Series[Sum[k^2 x^(k^2)/(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[DivisorSum[n, (-1)^(# + n/#) #^2 &, # <= Sqrt[n] &], {n, 1, 70}]

Formula

a(n) = Sum_{d|n, d <= sqrt(n)} (-1)^(d + n/d) * d^2.
Previous Showing 11-15 of 15 results.

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