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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A357051 a(n) = Sum_{d|n} 3^(n-d).

Original entry on oeis.org

1, 4, 10, 37, 82, 352, 730, 2998, 7291, 26488, 59050, 263170, 531442, 2127952, 5373460, 19669879, 43046722, 187086916, 387420490, 1607136634, 3878987860, 13947314752, 31381059610, 139902374692, 285916320883, 1129719740248, 2824682785300, 10460357985970
Offset: 1

Views

Author

Seiichi Manyama, Dec 20 2022

Keywords

Crossrefs

Cf. A074854, A112329, A359206.
Cf. A342628, A342629, A359203.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, 3^(n-#) &]; Array[a, 28] (* Amiram Eldar, Aug 23 2023 *)
  • PARI
    a(n) = sumdiv(n, d, 3^(n-d));
  • PARI
    my(N=30, x='x+O('x^N)); Vec(sum(k=1, N, 3^(k-1)*x^k/(1-3^(k-1)*x^k)))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(sum(k=1, N, x^k/(1-(3*x)^k)))

Formula

G.f.: Sum_{k>=1} 3^(k-1) * x^k/(1 - 3^(k-1) * x^k).
G.f.: Sum_{k>=1} x^k/(1 - (3 * x)^k).

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).

A359206 a(n) = Sum_{d|n} 4^(n-d).

Original entry on oeis.org

1, 5, 17, 81, 257, 1345, 4097, 20737, 69633, 328705, 1048577, 5574657, 16777217, 83902465, 286261249, 1359020033, 4294967297, 22565617665, 68719476737, 348967141377, 1168499539969, 5497562333185, 17592186044417, 93531519582209, 282574488338433
Offset: 1

Views

Author

Seiichi Manyama, Dec 20 2022

Keywords

Crossrefs

Cf. A074854, A112329, A357051.
Cf. A342628, A342629, A359204.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, 4^(n-#) &]; Array[a, 25] (* Amiram Eldar, Aug 23 2023 *)
  • PARI
    a(n) = sumdiv(n, d, 4^(n-d));
  • PARI
    my(N=30, x='x+O('x^N)); Vec(sum(k=1, N, 4^(k-1)*x^k/(1-4^(k-1)*x^k)))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(sum(k=1, N, x^k/(1-(4*x)^k)))

Formula

G.f.: Sum_{k>=1} 4^(k-1) * x^k/(1 - 4^(k-1) * x^k).
G.f.: Sum_{k>=1} x^k/(1 - (4 * x)^k).

A085010 a(n)=2^(2^n)*sum(k=0,n,1/2^(2^k)).

Original entry on oeis.org

1, 3, 13, 209, 53505, 3506503681, 15060318633198616577, 277813843495134114797235287762174738433, 94535152227927400227782074307303551040545228366095741656402842333161034088449
Offset: 0

Views

Author

Benoit Cloitre, Jun 19 2003

Keywords

Crossrefs

Cf. A007404.

Formula

a(n)=A074854(2^n)=; a(n)=floor(c*2^(2^n)) where c=sum(k>=0, 1/2^(2^k))=0.81642150902...
a(n + 1) = 1 + a(n)*2^(2^n), a(0) = 1 [From Peter Moxey (pmoxey(AT)live.com), Mar 14 2010]

A113978 a(n)=Sum(d|n)(10^(n-d)).

Original entry on oeis.org

1, 11, 101, 1101, 10001, 111001, 1000001, 11010001, 101000001, 1100100001, 10000000001, 111101000001, 1000000000001, 11000010000001, 101010000000001, 1101000100000001, 10000000000000001, 111001001000000001
Offset: 0

Views

Author

Paul Barry, Nov 11 2005

Keywords

Comments

A074854 in base 2. Stacking the elements gives the triangle A051731.

Programs

  • PARI
    a(n)=if(n<1,0,10^n*polcoeff(sum(k=1,n,10/(10-x^k),x*O(x^n)),n))

Formula

G.f.: Sum_{k>0} 10^(k-1)*x^k/(1-10^(k-1)*x^k).
Previous Showing 11-15 of 15 results.

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