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 1-5 of 5 results.

A300275 G.f.: 1 + Sum_{n>=1} a(n)*x^n/(1 - x^n) = Product_{n>=1} 1/(1 - x^n)^n.

Original entry on oeis.org

1, 2, 5, 10, 23, 40, 85, 147, 276, 474, 858, 1421, 2484, 4079, 6850, 11137, 18333, 29277, 47329, 74768, 118703, 185614, 290782, 449568, 696009, 1066258, 1632376, 2479057, 3759611, 5661568, 8512308, 12722132, 18974109, 28157619, 41690937, 61453929, 90379783
Offset: 1

Views

Author

Ilya Gutkovskiy, Mar 01 2018

Keywords

Comments

Moebius transform of A000219.
From Gus Wiseman, Jan 21 2019: (Start)
Also the number of plane partitions of n with relatively prime entries. For example, the a(4) = 10 plane partitions are:
31 211 1111
.
3 21 11 111
1 1 11 1
.
2 11
1 1
1 1
.
1
1
1
1
Also the number of plane partitions of n whose multiset of rows is aperiodic, meaning its multiplicities are relatively prime. For example, the a(4) = 10 plane partitions are:
4 31 22 211 1111
.
3 21 111
1 1 1
.
2 11
1 1
1 1
(End)

Links

Crossrefs

Cf. A000219, A000837, A078374, A300274, A300276, A300277, A300278.
Cf. A100953, A303546, A320802, A323584, A323585, A323587.

Programs

  • Maple
    with(numtheory):
b:= proc(n) option remember; `if`(n=0, 1, add(
      b(n-j)*sigma[2](j), j=1..n)/n)
    end:
a:= n-> add(b(d)*mobius(n/d), d=divisors(n)):
seq(a(n), n=1..40);  # Alois P. Heinz, Jun 21 2018
  • Mathematica
    nn = 37; f[x_] := 1 + Sum[a[n] x^n/(1 - x^n), {n, 1, nn}]; sol = SolveAlways[0 == Series[f[x] - Product[1/(1 - x^n)^n, {n, 1, nn}], {x, 0, nn}], x]; Table[a[n], {n, 1, nn}] /. sol // Flatten
s[n_] := SeriesCoefficient[Product[1/(1 - x^k)^k, {k, 1, n}], {x, 0, n}]; a[n_] := Sum[MoebiusMu[n/d] s[d], {d, Divisors[n]}]; Table[a[n], {n, 1, 37}]

Formula

a(n) = Sum_{d|n} mu(n/d)*A000219(d).

A300274 G.f.: 1 + Sum_{n>=1} a(n)*x^n/(1 - x^n) = Product_{n>=1} (1 + x^n)/(1 - x^n).

Original entry on oeis.org

2, 2, 6, 10, 22, 30, 62, 86, 146, 206, 342, 454, 726, 974, 1442, 1962, 2862, 3762, 5398, 7094, 9834, 12942, 17726, 22938, 31042, 40094, 53254, 68518, 90246, 114914, 150142, 190550, 245906, 310942, 398554, 500474, 637590, 797534, 1007714, 1255850, 1578526, 1956786
Offset: 1

Views

Author

Ilya Gutkovskiy, Mar 01 2018

Keywords

Comments

Moebius transform of A015128.

Links

Crossrefs

Cf. A000837, A015128, A078374, A300275, A300276, A300277, A300278.

Programs

  • Mathematica
    nn = 42; f[x_] := 1 + Sum[a[n] x^n/(1 - x^n), {n, 1, nn}]; sol = SolveAlways[0 == Series[f[x] - Product[(1 + x^n)/(1 - x^n), {n, 1, nn}], {x, 0, nn}], x]; Table[a[n], {n, 1, nn}] /. sol // Flatten
s[n_] := SeriesCoefficient[Product[(1 + x^k)/(1 - x^k), {k, 1, n}], {x, 0, n}]; a[n_] := Sum[MoebiusMu[n/d] s[d], {d, Divisors[n]}]; Table[a[n], {n, 1, 42}]

Formula

a(n) = Sum_{d|n} mu(n/d)*A015128(d).

A300276 G.f.: 1 + Sum_{n>=1} a(n)*x^n/(1 - x^n) = Product_{n>=1} (1 + x^n)^n.

Original entry on oeis.org

1, 1, 4, 6, 15, 22, 48, 75, 137, 218, 384, 593, 1003, 1549, 2501, 3857, 6110, 9256, 14408, 21675, 33081, 49422, 74483, 110135, 164116, 240955, 355027, 517553, 755893, 1093649, 1584518, 2277986, 3274887, 4679619, 6682635, 9491959, 13471238, 19030370, 26849913, 37734570
Offset: 1

Views

Author

Ilya Gutkovskiy, Mar 01 2018

Keywords

Comments

Moebius transform of A026007.

Links

Crossrefs

Cf. A000837, A026007, A078374, A300274, A300275, A300277, A300278.

Programs

  • Mathematica
    nn = 40; f[x_] := 1 + Sum[a[n] x^n/(1 - x^n), {n, 1, nn}]; sol = SolveAlways[0 == Series[f[x] - Product[(1 + x^n)^n, {n, 1, nn}], {x, 0, nn}], x]; Table[a[n], {n, 1, nn}] /. sol // Flatten
s[n_] := SeriesCoefficient[Product[(1 + x^k)^k, {k, 1, n}], {x, 0, n}]; a[n_] := Sum[MoebiusMu[n/d] s[d], {d, Divisors[n]}]; Table[a[n], {n, 1, 40}]

Formula

a(n) = Sum_{d|n} mu(n/d)*A026007(d).

A300278 G.f.: 1 + Sum_{n>=1} a(n)*x^n/(1 - x^n) = Product_{n>=1} (1 + n*x^n).

Original entry on oeis.org

1, 1, 4, 5, 14, 19, 42, 57, 115, 170, 287, 433, 694, 1061, 1709, 2461, 3740, 5635, 8243, 12256, 18255, 26135, 37826, 54209, 78315, 110488, 159418, 224514, 315414, 442790, 618665, 855640, 1199409, 1642334, 2288904, 3144738, 4303994, 5862294, 8031872, 10869290, 14749050
Offset: 1

Views

Author

Ilya Gutkovskiy, Mar 01 2018

Keywords

Comments

Moebius transform of A022629.

Links

Crossrefs

Cf. A000837, A022629, A078374, A085410, A300274, A300275, A300276, A300277.

Programs

  • Mathematica
    nn = 41; f[x_] := 1 + Sum[a[n] x^n/(1 - x^n), {n, 1, nn}]; sol = SolveAlways[0 == Series[f[x] - Product[(1 + n x^n), {n, 1, nn}], {x, 0, nn}], x]; Table[a[n], {n, 1, nn}] /. sol // Flatten
s[n_] := SeriesCoefficient[Product[(1 + k x^k), {k, 1, n}], {x, 0, n}]; a[n_] := Sum[MoebiusMu[n/d] s[d], {d, Divisors[n]}]; Table[a[n], {n, 1, 41}]

Formula

a(n) = Sum_{d|n} mu(n/d)*A022629(d).

A318025 Expansion of Sum_{k>=1} (-1 + Product_{j>=1} 1/(1 - j*x^(k*j))).

Original entry on oeis.org

1, 4, 7, 18, 26, 66, 98, 216, 361, 701, 1171, 2287, 3763, 6887, 11707, 20740, 34637, 60678, 100581, 172609, 285924, 481671, 791317, 1323831, 2156856, 3561119, 5784021, 9459559, 15250217, 24783964, 39713789, 64032664, 102200203, 163617694, 259745174, 413886941, 653715969, 1035539948
Offset: 1

Views

Author

Ilya Gutkovskiy, Aug 23 2018

Keywords

Comments

Inverse Moebius transform of A006906.

Links

Crossrefs

Cf. A006906, A047968, A300277, A302549, A318290.

Programs

  • Mathematica
    nmax = 38; Rest[CoefficientList[Series[Sum[-1 + Product[1/(1 - j x^(k j)), {j, 1, nmax}], {k, 1, nmax}], {x, 0, nmax}], x]]
b[n_] := b[n] = SeriesCoefficient[Product[1/(1 - k x^k), {k, 1, n}], {x, 0, n}]; a[n_] := a[n] = SeriesCoefficient[Sum[b[k] x^k/(1 - x^k), {k, 1, n}], {x, 0, n}]; Table[a[n], {n, 38}]
Table[Sum[Total[Times @@@ IntegerPartitions[d]], {d, Divisors[n]}], {n, 38}]

Formula

G.f.: Sum_{k>=1} A006906(k)*x^k/(1 - x^k).
a(n) = Sum_{d|n} A006906(d).
Showing 1-5 of 5 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.