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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A318368 a(n) = Sum_{d|n} (-1)^(n/d+1)*d*2^(d-1).

Original entry on oeis.org

1, 3, 13, 27, 81, 183, 449, 987, 2317, 5043, 11265, 24399, 53249, 114243, 245853, 523227, 1114113, 2357175, 4980737, 10480587, 22020557, 46126083, 96468993, 201302799, 419430481, 872361987, 1811941645, 3757981275, 7784628225, 16105886823, 33285996545, 68718951387, 141733932045
Offset: 1

Views

Author

Ilya Gutkovskiy, Aug 24 2018

Keywords

Crossrefs

Cf. A001787, A054599, A083413, A098407.

Programs

  • Mathematica
    Table[Sum[(-1)^(n/d + 1) d 2^(d - 1), {d, Divisors[n]}], {n, 33}]
nmax = 33; Rest[CoefficientList[Series[Sum[k 2^(k - 1) x^k/(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x]]
nmax = 33; Rest[CoefficientList[Series[Log[Product[(1 + x^k)^(2^(k - 1)), {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]]
  • PARI
    a(n) = sumdiv(n, d, (-1)^(n/d+1)*d*2^(d-1)); \\ Michel Marcus, Aug 25 2018

Formula

G.f.: Sum_{k>=1} k*2^(k-1)*x^k/(1 + x^k).
L.g.f.: log(Product_{k>=1} (1 + x^k)^(2^(k-1))) = Sum_{n>=1} a(n)*x^n/n.
a(n) ~ n * 2^(n-1). - Vaclav Kotesovec, Aug 25 2018

A359041 Number of finite sets of integer partitions with all equal sums and total sum n.

Original entry on oeis.org

1, 1, 2, 3, 6, 7, 14, 15, 32, 31, 63, 56, 142, 101, 240, 211, 467, 297, 985, 490, 1524, 1247, 2542, 1255, 6371, 1979, 7486, 7070, 14128, 4565, 32953, 6842, 42229, 37863, 56266, 17887, 192914, 21637, 145820, 197835, 371853, 44583, 772740, 63261, 943966, 1124840
Offset: 0

Views

Author

Gus Wiseman, Dec 14 2022

Keywords

Examples

			The a(1) = 1 through a(6) = 14 sets:
  {(1)}  {(2)}   {(3)}    {(4)}       {(5)}      {(6)}
         {(11)}  {(21)}   {(22)}      {(32)}     {(33)}
                 {(111)}  {(31)}      {(41)}     {(42)}
                          {(211)}     {(221)}    {(51)}
                          {(1111)}    {(311)}    {(222)}
                          {(2),(11)}  {(2111)}   {(321)}
                                      {(11111)}  {(411)}
                                                 {(2211)}
                                                 {(3111)}
                                                 {(21111)}
                                                 {(111111)}
                                                 {(3),(21)}
                                                 {(3),(111)}
                                                 {(21),(111)}

Crossrefs

This is the constant-sum case of A261049, ordered A358906.
The version for all different sums is A271619, ordered A336342.
Allowing repetition gives A305551, ordered A279787.
The version for compositions instead of partitions is A358904.
A001970 counts multisets of partitions.
A034691 counts multisets of compositions, ordered A133494.
A098407 counts sets of compositions, ordered A358907.
Cf. A000005, A000041, A038041, A055887, A063834, A074854, A289078, A304961, A305552, A306017.

Programs

  • Mathematica
    Table[If[n==0,1,Sum[Binomial[PartitionsP[d],n/d],{d,Divisors[n]}]],{n,0,50}]
  • PARI
    a(n) = if (n, sumdiv(n, d, binomial(numbpart(d), n/d)), 1); \\ Michel Marcus, Dec 14 2022

Formula

a(n) = Sum_{d|n} binomial(A000041(d),n/d).

A343331 Expansion of Product_{k>=1} (1 + x^k)^(10^(k-1)).

Original entry on oeis.org

1, 1, 10, 110, 1145, 12045, 126070, 1319570, 13798710, 144217910, 1506406702, 15726571002, 164096557935, 1711386871635, 17839701265570, 185876723016390, 1935830424374840, 20152131324766520, 209696974024339610, 2181155691766631710, 22678274833738085501, 235704268837407670401
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 12 2021

Keywords

Comments

In general, if m > 1 and g.f. = Product_{k>=1} (1 + x^k)^(m^(k-1)), then a(n, m) ~ exp(2*sqrt(n/m) - 1/(2*m) - c(m)/m) * m^(n - 1/4) / (2*sqrt(Pi)*n^(3/4)), where c(m) = Sum_{j>=2} (-1)^j / (j * (m^(j-1) - 1)). - Vaclav Kotesovec, Apr 13 2021

Crossrefs

Cf. A098407, A292844, A343355, A343360, A343361, A343362, A343363, A343364, A343365, A343366.

Programs

  • Maple
    h:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(h(n-i*j, i-1)*binomial(10^(i-1), j), j=0..n/i)))
    end:
a:= n-> h(n$2):
seq(a(n), n=0..21);  # Alois P. Heinz, Apr 12 2021
  • Mathematica
    nmax = 21; CoefficientList[Series[Product[(1 + x^k)^(10^(k - 1)), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, (1/n) Sum[Sum[(-1)^(k/d + 1) d 10^(d - 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 21}]

Formula

a(n) ~ exp(sqrt(2*n/5) - 1/20 - c/10) * 10^(n - 1/4) / (2*sqrt(Pi)*n^(3/4)), where c = Sum_{j>=2} (-1)^j / (j * (10^(j-1) - 1)). - Vaclav Kotesovec, Apr 13 2021

A371484 Expansion of e.g.f. Product_{k>=1} (1 + x^k/k!)^(2^(k-1)).

Original entry on oeis.org

1, 1, 2, 10, 30, 166, 968, 5440, 39112, 289096, 2330832, 19947280, 185603520, 1807683008, 18554569504, 203803276960, 2315043684112, 27640348895248, 344414508782624, 4483617418231456, 60518677509951520, 846362835642437536, 12289340439560875264
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 25 2024

Keywords

Comments

"EGJ" (unordered, element, labeled) transform of A011782.

Links

Crossrefs

Cf. A007837, A011782, A032315, A032316, A098407, A371482.

Programs

  • Mathematica
    nmax = 22; CoefficientList[Series[Product[(1 + x^k/k!)^(2^(k - 1)), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
Previous Showing 21-24 of 24 results.

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