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 1-2 of 2 results.

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).

A335227 G.f.: x / (Sum_{k>=1} k * x^k / (1 + x^k)).

Original entry on oeis.org

1, -1, -3, 6, 1, -20, 24, 38, -132, 34, 411, -632, -601, 2914, -1664, -7822, 15649, 6802, -62082, 55672, 141109, -369310, -12036, 1275642, -1580834, -2343886, 8375349, -2648282, -25217490, 41097852, 33815048, -183252284, 117569579, 475949186, -1006346968, -344955964
Offset: 0

Views

Author

Ilya Gutkovskiy, May 27 2020

Keywords

Crossrefs

Cf. A000009, A000593, A002039, A180305, A320651, A335228.

Programs

  • Mathematica
    nmax = 35; CoefficientList[Series[x/Sum[k x^k/(1 + x^k), {k, 1, nmax + 1}], {x, 0, nmax}], x]
nmax = 35; CoefficientList[Series[1/D[Log[Product[(1 + x^k), {k, 1, nmax + 1}]], x], {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = -Sum[DivisorSum[k + 1, # &, OddQ[#] &] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 35}]

Formula

G.f.: x / (Sum_{k>=1} (-1)^(k+1) * x^k / (1 - x^k)^2).
G.f.: 1 / log(g(x))', where g(x) = Product_{k>=1} (1 + x^k) is the g.f. for A000009.
G.f.: 1 / (Sum_{k>=0} A000593(k+1) * x^k).
a(0) = 1; a(n) = -Sum_{k=1..n} A000593(k+1) * a(n-k).
Showing 1-2 of 2 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.