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.

Previous Showing 31-35 of 35 results.

A307574 Expansion of Product_{k>=1} (1 - (x/(1-x))^k)^k.

Original entry on oeis.org

1, -1, -3, -6, -10, -11, 3, 63, 240, 677, 1622, 3415, 6277, 9485, 8917, -9299, -83683, -309568, -902995, -2315518, -5411355, -11662530, -23117627, -41317787, -62820880, -65358588, 29550902, 449154266, 1783671567, 5453429052, 14668699694, 36273441659
Offset: 0

Views

Author

Seiichi Manyama, Apr 15 2019

Keywords

Crossrefs

Convolution inverse of A320563.
Cf. A073592, A307310, A320564.

Programs

  • Mathematica
    m = 31; CoefficientList[Series[Product[(1 - (x/(1-x))^k)^k, {k, 1, m}], {x, 0, m}], x] (* Amiram Eldar, May 14 2021 *)
  • PARI
    N=66; x='x+O('x^N); Vec(prod(k=1, N, (1-(x/(1-x))^k)^k))

A319108 Expansion of Product_{k>=1} (1 - x^k)^(k-1).

Original entry on oeis.org

1, 0, -1, -2, -3, -2, -1, 4, 8, 14, 17, 18, 9, -4, -27, -58, -88, -114, -122, -106, -48, 48, 200, 376, 577, 746, 862, 840, 646, 208, -486, -1450, -2622, -3888, -5086, -5950, -6204, -5492, -3547, -44, 5036, 11732, 19582, 28034, 35932, 42042, 44519, 41660, 31450, 12382, -16721
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 10 2018

Keywords

Comments

Convolution of A000041 and A073592.
Convolution inverse of A052847.

Crossrefs

Cf. A000041, A000203, A001157, A052847, A073592, A299019.

Programs

  • Maple
    a:=series(mul((1-x^k)^(k-1),k=1..100),x=0,51): seq(coeff(a,x,n),n=0..50); # Paolo P. Lava, Apr 02 2019
  • Mathematica
    nmax = 50; CoefficientList[Series[Product[(1 - x^k)^(k - 1), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 50; CoefficientList[Series[Exp[Sum[(DivisorSigma[1, k] - DivisorSigma[2, k]) x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d (1 - d), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 50}]

Formula

G.f.: exp(Sum_{k>=1} (sigma_1(k) - sigma_2(k))*x^k/k), where sigma_1(k) = sum of divisors of k (A000203) and sigma_2(k) = sum of squares of divisors of k (A001157).

A319669 Expansion of Product_{k>=1} (1 - x^k)^(2*k-1).

Original entry on oeis.org

1, -1, -3, -2, 1, 10, 13, 15, -1, -30, -63, -89, -80, -14, 131, 304, 493, 561, 434, -32, -836, -1895, -2960, -3583, -3240, -1338, 2401, 8004, 14499, 20494, 23369, 20401, 8567, -13741, -46408, -85717, -124027, -149612, -147167, -101002, 2520, 168026, 388077, 634914
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 25 2018

Keywords

Crossrefs

Cf. A000203, A001157, A073592, A253289, A255835, A276551, A278945, A285069.

Programs

  • Maple
    a:=series(mul((1-x^k)^(2*k-1),k=1..100),x=0,44): seq(coeff(a,x,n),n=0..43); # Paolo P. Lava, Apr 02 2019
  • Mathematica
    nmax = 43; CoefficientList[Series[Product[(1 - x^k)^(2 k - 1), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 43; CoefficientList[Series[Exp[Sum[(DivisorSigma[1, k] - 2 DivisorSigma[2, k]) x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d (1 - 2 d), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 43}]

Formula

G.f.: exp(Sum_{k>=1} (sigma_1(k) - 2*sigma_2(k))*x^k/k).

A369726 Maximal coefficient of (1 - x) * (1 - x^2)^2 * (1 - x^3)^3 * ... * (1 - x^n)^n.

Original entry on oeis.org

1, 1, 2, 5, 30, 289, 5170, 155768, 7947236, 695357704, 105014923458, 26823702973095, 12124672181643014, 9302296598744837059, 12142590791028740988194, 26874517085010633423659616, 100413718348008543669377307304, 634527279123990475683817934978079
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 30 2024

Keywords

Crossrefs

Cf. A073592, A086376, A369710, A369711, A369712.

Programs

  • Maple
    a:= n-> max(coeffs(expand(mul((1-x^k)^k, k=1..n)))):
seq(a(n), n=0..17);  # Alois P. Heinz, Jan 30 2024
  • Mathematica
    Table[Max[CoefficientList[Product[(1 - x^k)^k, {k, 1, n}], x]], {n, 0, 17}]
  • PARI
    a(n) = vecmax(Vec(prod(k=1, n, (1-x^k)^k))); \\ Michel Marcus, Jan 30 2024

A381011 a(n) = [(x*y)^n] Product_{k>=1} (1 - x^k - y^k)^k.

Original entry on oeis.org

1, 0, 2, -6, -14, -10, 32, 76, -80, -340, -200, 590, 2302, 1890, -3470, -11468, -16254, 5244, 57406, 109340, 81396, -158664, -550388, -829558, -359856, 1509570, 4333256, 6198660, 2628406, -10133230, -30439512, -46214582, -29696680, 45589368
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 10 2025

Keywords

Crossrefs

Cf. A073592, A322213, A322214.

Programs

  • Mathematica
    Table[SeriesCoefficient[Product[(1 - x^k - y^k)^k, {k, 1, n}], {x, 0, n}, {y, 0, n}], {n, 0, 33}]
Previous Showing 31-35 of 35 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.