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 1-8 of 8 results.

A300277 G.f.: 1 + Sum_{n>=1} a(n)*x^n/(1 - x^n) = Product_{n>=1} 1/(1 - n*x^n).

Original entry on oeis.org

1, 2, 5, 11, 24, 48, 96, 184, 348, 645, 1169, 2140, 3761, 6687, 11645, 20326, 34635, 59854, 100579, 171211, 285718, 479325, 791315, 1318955, 2156805, 3553589, 5783306, 9445861, 15250215, 24759156, 39713787, 63991400, 102197851, 163548416, 259744930, 413761633, 653715967
Offset: 1

Views

Author

Ilya Gutkovskiy, Mar 01 2018

Keywords

Comments

Moebius transform of A006906.

Links

Crossrefs

Cf. A000837, A006906, A078374, A085410, A300274, A300275, A300276, A300278.

Programs

  • 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 - n x^n), {n, 1, nn}], {x, 0, nn}], x]; Table[a[n], {n, 1, nn}] /. sol // Flatten
s[n_] := SeriesCoefficient[Product[1/(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, 37}]

Formula

a(n) = Sum_{d|n} mu(n/d)*A006906(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).

A323584 Second Moebius transform of A000219. Number of plane partitions of n whose multiset of rows is aperiodic and whose multiset of columns is also aperiodic.

Original entry on oeis.org

1, 1, 1, 4, 8, 22, 34, 84, 137, 271, 450, 857, 1373, 2483, 3993, 6823, 10990, 18332, 28966, 47328, 74286, 118614, 184755, 290781, 448010, 695986, 1063773, 1632100, 2474970, 3759610, 5654233, 8512307, 12710995, 18973247, 28139285, 41690830, 61423271, 90379782
Offset: 0

Views

Author

Gus Wiseman, Jan 19 2019

Keywords

Comments

A multiset is aperiodic if its multiplicities are relatively prime.
Also the number of plane partitions of n whose multiset of rows is aperiodic and whose parts are relatively prime.

Examples

			The a(4) = 8 plane partitions with aperiodic multisets of rows and columns:
  4   31   211
.
  3   21   111
  1   1    1
.
  2   11
  1   1
  1   1
The a(4) = 8 plane partitions with aperiodic multiset of rows and relatively prime parts:
  31   211   1111
.
  3   21   111
  1   1    1
.
  2   11
  1   1
  1   1

Links

Crossrefs

Cf. A000219, A000837, A003293, A100953, A300275, A303546, A320802, A321390, A323585, A323587.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
ptnplane[n_]:=Union[Map[Reverse@*primeMS,Join@@Permutations/@facs[n],{2}]];
Table[Sum[Length[Select[ptnplane[Times@@Prime/@y],And[GCD@@Length/@Split[#]==1,And@@GreaterEqual@@@#,And@@(GreaterEqual@@@Transpose[PadRight[#]])]&]],{y,Select[IntegerPartitions[n],GCD@@#==1&]}],{n,10}]

Formula

The Moebius transform T of a sequence q is T(q)(n) = Sum_{d|n} mu(n/d) * q(d) where mu = A008683. The first Moebius transform of A000219 is A300275 and the third is A323585.

A323585 Third Moebius transform of A000219. Number of plane partitions of n whose multiset of rows is aperiodic and whose multiset of columns is also aperiodic and whose parts are relatively prime.

Original entry on oeis.org

1, 1, 0, 3, 7, 21, 30, 83, 129, 267, 428, 856, 1332, 2482, 3909, 6798, 10853, 18331, 28665, 47327, 73829, 118527, 183898, 290780, 446508, 695964, 1061290, 1631829, 2470970, 3759609, 5646952, 8512306, 12700005, 18972387, 28120953, 41690725, 61392966, 90379781
Offset: 0

Views

Author

Gus Wiseman, Jan 19 2019

Keywords

Comments

A multiset is aperiodic if its multiplicities are relatively prime.

Examples

			The a(4) = 7 plane partitions with aperiodic multisets of rows and columns and relatively prime parts:
  31   211
.
  3   21   111
  1   1    1
.
  2   11
  1   1
  1   1
The same for a(5) = 21:
  41   32   311   221   2111
.
  4   3   31   21   22   21   211   111   1111
  1   2   1    2    1    11   1     11    1
.
  3   2   21   11   111
  1   2   1    11   1
  1   1   1    1    1
.
  2   11
  1   1
  1   1
  1   1

Links

Crossrefs

Cf. A000219, A000837, A003293, A100953, A300275, A303546, A320802, A321390, A323584, A323587.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
ptnplane[n_]:=Union[Map[Reverse@*primeMS,Join@@Permutations/@facs[n],{2}]];
Table[Sum[Length[Select[ptnplane[Times@@Prime/@y],And[GCD@@Length/@Split[#]==1,GCD@@Length/@Split[Transpose[PadRight[#]]]==1,And@@GreaterEqual@@@#,And@@(GreaterEqual@@@Transpose[PadRight[#]])]&]],{y,Select[IntegerPartitions[n],GCD@@#==1&]}],{n,10}]

Formula

The Moebius transform T of a sequence q is T(q)(n) = Sum_{d|n} mu(n/d) * q(d) where mu = A008683. The first Moebius transform of A000219 is A300275 and the second is A323584.

A323587 Number of strict (distinct parts) plane partitions of n with relatively prime parts.

Original entry on oeis.org

1, 1, 0, 2, 2, 4, 6, 10, 12, 18, 30, 40, 48, 74, 92, 142, 172, 242, 294, 412, 490, 722, 854, 1164, 1396, 1880, 2260, 2976, 3748, 4764, 5792, 7472, 9082, 11488, 14012, 17522, 21830, 26896, 32820, 40536, 49488, 60636, 73626, 89962, 108854, 134240, 160952, 195858
Offset: 0

Views

Author

Gus Wiseman, Jan 20 2019

Keywords

Examples

			The a(9) = 18 plane partitions:
  81   72   621   54   531   432
.
  8   7   61   62   5   51   53   42   43
  1   2   2    1    4   3    1    3    2
.
  6   5   4
  2   3   3
  1   1   2

Crossrefs

Cf. A000219, A000837, A003293, A006951, A026007, A100883, A117433 (strict plane partitions), A300275 (plane partitions with relatively prime parts), A303546, A320802, A323584, A323585.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
ptnplane[n_]:=Union[Map[Reverse@*primeMS,Join@@Permutations/@facs[n],{2}]];
Table[Sum[Length[Select[ptnplane[Times@@Prime/@y],And[And@@GreaterEqual@@@#,And@@(GreaterEqual@@@Transpose[PadRight[#]])]&]],{y,Select[IntegerPartitions[n],UnsameQ@@#&&GCD@@#==1&]}],{n,30}]

Formula

Moebius transform of A117433.

A302549 Expansion of Sum_{k>=1} (-1 + Product_{j>=1} 1/(1 - x^(k*j))^j).

Original entry on oeis.org

1, 4, 7, 17, 25, 58, 87, 177, 289, 528, 860, 1550, 2486, 4257, 6910, 11474, 18335, 29941, 47331, 75819, 118887, 187338, 290784, 452904, 696058, 1071234, 1632947, 2487504, 3759613, 5676424, 8512310, 12744903, 18975839, 28194293, 41691157, 61516394, 90379785
Offset: 1

Views

Author

Ilya Gutkovskiy, Jun 20 2018

Keywords

Comments

Inverse Moebius transform of A000219.

Links

Crossrefs

Cf. A000219, A047966, A047968, A300275, A302550.

Programs

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

Formula

G.f.: Sum_{k>=1} A000219(k)*x^k/(1 - x^k).
a(n) = Sum_{d|n} A000219(d).
Showing 1-8 of 8 results.

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