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 1-3 of 3 results.

A321261 a(n) = [x^n] Product_{k>=1} (1 + x^k)^(sigma_n(k)-k^n).

Original entry on oeis.org

1, 0, 1, 1, 17, 2, 859, 131, 105508, 40907, 72916903, 6834168, 228239366293, 27616985835, 2050004858009336, 352807044193881, 87173272463714343166, 6798224808203572198, 18318379579349549499397403, 1187836799227050499295342, 11258903016282277676462826232428
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 01 2018

Keywords

Crossrefs

Cf. A281268, A318844, A319107, A321042, A321258, A321260.

Programs

  • Mathematica
    Table[SeriesCoefficient[Product[(1 + x^k)^(DivisorSigma[n, k] - k^n), {k, 1, n}], {x, 0, n}], {n, 0, 20}]
Table[SeriesCoefficient[Exp[Sum[Sum[(-1)^(k/d + 1) d (DivisorSigma[n, d] - d^n), {d, Divisors[k]}] x^k/k, {k, 1, n}]], {x, 0, n}], {n, 0, 20}]

Formula

a(n) = [x^n] exp(Sum_{k>=1} ( Sum_{d|k} (-1)^(k/d+1)*d*(sigma_n(d) - d^n) ) * x^k/k).

A326831 Expansion of Product_{i>=2, j>=2} (1 + x^(i*j))^j.

Original entry on oeis.org

1, 0, 0, 0, 2, 0, 5, 0, 7, 3, 17, 0, 37, 6, 58, 23, 120, 21, 235, 67, 390, 161, 726, 230, 1349, 521, 2225, 1055, 3990, 1714, 7040, 3341, 11604, 6294, 20053, 10500, 34252, 19115, 56055, 34168, 94306, 56998, 157078, 99515, 254766, 171484, 419287, 283565
Offset: 0

Views

Author

Ilya Gutkovskiy, Oct 20 2019

Keywords

Comments

Weigh transform of A048050.
Convolution of A326831 and A025147 is A319107. - Vaclav Kotesovec, Oct 26 2019

Links

Crossrefs

Cf. A048050, A192065, A319107, A326830.

Programs

  • Maple
    with(numtheory):
g:= proc(n) option remember; `if`(n<4, 0, sigma(n)-1-n) end:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1)*binomial(g(i), j), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..50);  # Alois P. Heinz, Oct 20 2019
  • Mathematica
    nmax = 47; CoefficientList[Series[Product[(1 + x^k)^(DivisorSigma[1, k] - k - 1), {k, 2, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[If[d == 1, 0, (-1)^(k/d + 1) d (DivisorSigma[1, d] - d - 1)], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 47}]

Formula

G.f.: Product_{k>=1} (1 + x^k)^A048050(k).
a(n) ~ exp(3*(2*(Pi^2 - 6)*Zeta(3))^(1/3) * n^(2/3)/4 - Pi^2 * n^(1/3) / (2^(7/3) * ((Pi^2 - 6)*Zeta(3))^(1/3)) - Pi^4 / (96*(Pi^2 - 6)*Zeta(3))) * 2^(19/24) * ((Pi^2 - 6)*Zeta(3))^(1/6) / (sqrt(3*Pi) * n^(2/3)). - Vaclav Kotesovec, Oct 26 2019

A321262 Expansion of 1/(1 - Sum_{k>=1} k*x^(2*k)/(1 - x^k)).

Original entry on oeis.org

1, 0, 1, 1, 4, 3, 14, 12, 43, 50, 140, 177, 474, 643, 1560, 2325, 5246, 8194, 17763, 28838, 60190, 101063, 204935, 352227, 700037, 1224816, 2394971, 4250616, 8209174, 14724570, 28175997, 50949079, 96797183, 176131780, 332804667, 608449008, 1144920041, 2100793404
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 01 2018

Keywords

Comments

Invert transform of A001065.

Links

Crossrefs

Cf. A001065, A180305, A318784, A319107, A320651, A321263.

Programs

  • Maple
    a:=series(1/(1-add(k*x^(2*k)/(1-x^k),k=1..100)),x=0,38): seq(coeff(a,x,n),n=0..37); # Paolo P. Lava, Apr 02 2019
  • Mathematica
    nmax = 37; CoefficientList[Series[1/(1 - Sum[k x^(2 k)/(1 - x^k), {k, 1, nmax}]), {x, 0, nmax}], x]
nmax = 37; CoefficientList[Series[1/(1 - Sum[(k - EulerPhi[k]) x^k/(1 - x^k), {k, 1, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[(DivisorSigma[1, k] - k) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 37}]

Formula

G.f.: 1/(1 - Sum_{k>=1} (sigma(k) - k)*x^k).
G.f.: 1/(1 - Sum_{k>=1} (k - phi(k))*x^k/(1 - x^k)).
a(0) = 1; a(n) = Sum_{k=1..n} A001065(k)*a(n-k).
Showing 1-3 of 3 results.

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