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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A343366 Expansion of Product_{k>=1} (1 + x^k)^(9^(k-1)).

Original entry on oeis.org

1, 1, 9, 90, 846, 8055, 76224, 721389, 6819192, 64422126, 608173020, 5737815756, 54100140735, 509794737636, 4801164836634, 45192001954005, 425156458320783, 3997756503852489, 37572655020653089, 352957677187938076, 3314174696310855888, 31105460092251410001, 291818245344169918725
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, A292843, A343354, A343360, A343361, A343362, A343363, A343364, A343365.

Programs

  • Maple
    h:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(h(n-i*j, i-1)*binomial(9^(i-1), j), j=0..n/i)))
    end:
a:= n-> h(n$2):
seq(a(n), n=0..22);  # Alois P. Heinz, Apr 12 2021
  • Mathematica
    nmax = 22; CoefficientList[Series[Product[(1 + x^k)^(9^(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 9^(d - 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 22}]
  • PARI
    seq(n)={Vec(prod(k=1, n, (1 + x^k + O(x*x^n))^(9^(k-1))))} \\ Andrew Howroyd, Apr 12 2021

Formula

a(n) ~ exp(2*sqrt(n/9) - 1/18 - c/9) * 9^(n - 1/4) / (2*sqrt(Pi)*n^(3/4)), where c = Sum_{j>=2} (-1)^j / (j * (9^(j-1) - 1)). - Vaclav Kotesovec, Apr 13 2021

A299023 Number of compositions of n whose standard factorization into Lyndon words has all strict compositions as factors.

Original entry on oeis.org

1, 2, 4, 7, 12, 23, 38, 66, 112, 193, 319, 539, 887, 1466, 2415, 3951, 6417, 10428, 16817, 27072, 43505, 69560, 110916, 176469, 279893, 442742, 698919, 1100898, 1729530, 2712134, 4244263, 6628174, 10332499, 16077835, 24972415, 38729239, 59958797, 92685287
Offset: 1

Views

Author

Gus Wiseman, Jan 31 2018

Keywords

Examples

			The a(5) = 12 compositions:
      (5) = (5)
     (41) = (4)*(1)
     (14) = (14)
     (32) = (3)*(2)
     (23) = (23)
    (311) = (3)*(1)*(1)
    (131) = (13)*(1)
    (221) = (2)*(2)*(1)
    (212) = (2)*(12)
   (2111) = (2)*(1)*(1)*(1)
   (1211) = (12)*(1)*(1)
  (11111) = (1)*(1)*(1)*(1)*(1)
Not included:
    (113) = (113)
    (122) = (122)
   (1121) = (112)*(1)
   (1112) = (1112)

Links

Crossrefs

Cf. A001045, A032020, A032153, A034691, A049311, A059966, A089259, A098407, A116540, A185700, A270995, A296373, A299024, A299026, A299027.

Programs

  • Mathematica
    nn=50;
ser=Product[1/(1-x^n)^Total[(Length[#]-1)!&/@Select[IntegerPartitions[n],UnsameQ@@#&]],{n,nn}];
Table[SeriesCoefficient[ser,{x,0,n}],{n,nn}]
  • PARI
    EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
seq(N)={EulerT(Vec(sum(n=1, N-1, (n-1)!*x^(n*(n+1)/2)/prod(k=1, n, 1-x^k + O(x^N)))))} \\ Andrew Howroyd, Dec 01 2018

Formula

Euler transform of A032153.

A299024 Number of compositions of n whose standard factorization into Lyndon words has distinct strict compositions as factors.

Original entry on oeis.org

1, 1, 3, 4, 7, 13, 21, 34, 58, 98, 158, 258, 421, 676, 1108, 1777, 2836, 4544, 7220, 11443, 18215, 28729, 45203, 71139, 111518, 174402, 272367, 424892, 660563, 1025717, 1590448, 2460346, 3800816, 5862640, 9026963, 13885425, 21321663, 32695098, 50073855
Offset: 1

Views

Author

Gus Wiseman, Jan 31 2018

Keywords

Examples

			The a(5) = 7 compositions:
      (5) = (5)
     (41) = (4)*(1)
     (14) = (14)
     (32) = (3)*(2)
     (23) = (23)
    (131) = (13)*(1)
    (212) = (2)*(12)
Not included:
    (311) = (3)*(1)*(1)
    (113) = (113)
    (221) = (2)*(2)*(1)
    (122) = (122)
   (2111) = (2)*(1)*(1)*(1)
   (1211) = (12)*(1)*(1)
   (1121) = (112)*(1)
   (1112) = (1112)
  (11111) = (1)*(1)*(1)*(1)*(1)

Links

Crossrefs

Cf. A001045, A032020, A032153, A034691, A049311, A050342, A059966, A089259, A098407, A116540, A185700, A270995, A283877, A292432, A296373, A299023, A299026, A299027.

Programs

  • Mathematica
    nn=50;
ser=Product[(1+x^n)^Total[(Length[#]-1)!&/@Select[IntegerPartitions[n],UnsameQ@@#&]],{n,nn}];
Table[SeriesCoefficient[ser,{x,0,n}],{n,nn}]
  • PARI
    WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v,n,(-1)^(n-1)/n))))-1,-#v)}
seq(N)={WeighT(Vec(sum(n=1, N-1, (n-1)!*x^(n*(n+1)/2)/prod(k=1, n, 1-x^k + O(x^N)))))} \\ Andrew Howroyd, Dec 01 2018

Formula

Weigh transform of A032153.

A299026 Number of compositions of n whose standard factorization into Lyndon words has all weakly increasing factors.

Original entry on oeis.org

1, 2, 4, 8, 16, 31, 59, 111, 205, 378, 685, 1238, 2213, 3940, 6955, 12221, 21333, 37074, 64073, 110267, 188877, 322244, 547522, 926903, 1563370, 2628008, 4402927, 7353656, 12244434, 20329271, 33657560, 55574996, 91525882, 150356718, 246403694, 402861907
Offset: 1

Views

Author

Gus Wiseman, Feb 01 2018

Keywords

Examples

			The 2^6 - a(7) = 5 compositions of 7 whose Lyndon prime factors are not all weakly increasing: (11212), (1132), (1213), (1321), (142).

Links

Crossrefs

Cf. A001045, A032153, A034691, A049311, A059966, A098407, A167934, A185700, A270995, A296373, A299023, A299024, A299027.

Programs

  • Mathematica
    nn=50;
ser=Product[1/(1-x^n)^(PartitionsP[n]-DivisorSigma[0,n]+1),{n,nn}];
Table[SeriesCoefficient[ser,{x,0,n}],{n,nn}]
  • PARI
    EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
seq(n)={EulerT(vector(n, n, numbpart(n) - numdiv(n) + 1))} \\ Andrew Howroyd, Dec 01 2018

Formula

Euler transform of A167934.

A299027 Number of compositions of n whose standard factorization into Lyndon words has all distinct weakly increasing factors.

Original entry on oeis.org

1, 1, 3, 5, 11, 20, 38, 69, 125, 225, 400, 708, 1244, 2176, 3779, 6532, 11229, 19223, 32745, 55555, 93875, 158025, 265038, 443009, 738026, 1225649, 2029305, 3350167, 5515384, 9055678, 14830076, 24226115, 39480306, 64190026, 104130753, 168556588, 272268482
Offset: 1

Views

Author

Gus Wiseman, Feb 01 2018

Keywords

Examples

			The a(5) = 11 compositions:
      (5) = (5)
     (41) = (4)*(1)
     (14) = (14)
     (32) = (3)*(2)
     (23) = (23)
    (131) = (13)*(1)
    (113) = (113)
    (212) = (2)*(12)
    (122) = (122)
   (1121) = (112)*(1)
   (1112) = (1112)
Not included:
    (311) = (3)*(1)*(1)
    (221) = (2)*(2)*(1)
   (2111) = (2)*(1)*(1)*(1)
   (1211) = (12)*(1)*(1)
  (11111) = (1)*(1)*(1)*(1)*(1)

Links

Crossrefs

Cf. A001045, A032153, A034691, A049311, A059966, A098407, A116540, A185700, A270995, A283877, A292432, A293993, A296373, A299023, A299024, A299026.

Programs

  • Mathematica
    nn=50;
ser=Product[(1+x^n)^(PartitionsP[n]-DivisorSigma[0,n]+1),{n,nn}];
Table[SeriesCoefficient[ser,{x,0,n}],{n,nn}]
  • PARI
    WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v,n,(-1)^(n-1)/n))))-1,-#v)}
seq(n)={WeighT(vector(n, n, numbpart(n) - numdiv(n) + 1))} \\ Andrew Howroyd, Dec 01 2018

Formula

Weigh transform of A167934.

A358904 Number of finite sets of compositions with all equal sums and total sum n.

Original entry on oeis.org

1, 1, 2, 4, 9, 16, 38, 64, 156, 260, 632, 1024, 2601, 4096, 10208, 16944, 40966, 65536, 168672, 262144, 656980, 1090240, 2620928, 4194304, 10862100, 16781584, 41940992, 69872384, 168403448, 268435456, 693528552, 1073741824, 2695006177, 4473400320, 10737385472
Offset: 0

Views

Author

Gus Wiseman, Dec 13 2022

Keywords

Examples

			The a(1) = 1 through a(4) = 9 sets:
  {(1)}  {(2)}   {(3)}    {(4)}
         {(11)}  {(12)}   {(13)}
                 {(21)}   {(22)}
                 {(111)}  {(31)}
                          {(112)}
                          {(121)}
                          {(211)}
                          {(1111)}
                          {(2),(11)}

Crossrefs

This is the constant-sum case of A098407, ordered A358907.
The version for distinct sums is A304961, ordered A336127.
Allowing repetition gives A305552, ordered A074854.
The case of sets of partitions is A359041.
A001970 counts multisets of partitions.
A034691 counts multisets of compositions, ordered A133494.
A261049 counts sets of partitions, ordered A358906.
Cf. A000009, A063834, A075900, A218482, A296122.

Programs

  • Mathematica
    Table[If[n==0,1,Sum[Binomial[2^(d-1),n/d],{d,Divisors[n]}]],{n,0,30}]
  • PARI
    a(n) = if (n, sumdiv(n, d, binomial(2^(d-1), n/d)), 1); \\ Michel Marcus, Dec 14 2022

Formula

a(n>0) = Sum_{d|n} binomial(2^(d-1),n/d).

A304962 Expansion of Product_{k>=1} ((1 + x^k)/(1 - x^k))^(2^(k-1)).

Original entry on oeis.org

1, 2, 6, 18, 50, 138, 374, 994, 2610, 6778, 17414, 44346, 112034, 280970, 700038, 1733706, 4269970, 10463154, 25518198, 61962458, 149839602, 360958306, 866405702, 2072579058, 4942074082, 11748730482, 27849974598, 65837539522, 155236876018, 365125130490, 856767548022
Offset: 0

Views

Author

Ilya Gutkovskiy, May 22 2018

Keywords

Comments

Convolution of the sequences A034691 and A098407.

Links

Crossrefs

Cf. A011782, A015128, A034691, A098407, A156616, A261519, A302239.

Programs

  • Maple
    g:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      2^(d-1), d=numtheory[divisors](j))*g(n-j), j=1..n)/n)
    end:
b:= proc(n, i) option remember; `if`(n=0 or i=1, `if`(n>1, 0, 1),
      add(b(n-i*j, i-1)*binomial(2^(i-1), j), j=0..n/i))
    end:
a:= n-> add(g(n-j)*b(j$2), j=0..n):
seq(a(n), n=0..35);  # Alois P. Heinz, May 22 2018
# Maple program to compute c(n) from a(n) or a(n) from c(n).
with(numtheory):
andrews:=proc(liste) local n,z,serie,ls,i,d,aaa;
   n:=nops(liste);
aaa:=liste;
serie:=listtoseries(aaa,z,ogf):
ls:=series(ln(serie),z,n);
   [seq(coeff(ls,z,d),d=1..n)];
   [seq(elemmobius(%,i),i=1..n-1)]
end:
swerdna:=proc(liste) local n,i,z;
  n:=nops(liste);
  series(convert([seq((1-z^i)^(-liste[i]),i=1..n)],`*`),z,n);
  [seq(coeff(%,z,i),i=0..n-1)]
end:
elemmobius:=proc(liste,d) local k,rep;
   rep:=0;
   for k in divisors(d) do
      rep:=rep+liste[k]*mobius(iquo(d,k))/iquo(d,k)
   od;
   rep
end:
# Here andrews() finds the c(n) and swerdna() finds the a(n) if the c(n) are known.
# For ordinary partitions the c(n) are [1,1,1,1,1, ...].
# Simon Plouffe, Jun 20 2018
  • Mathematica
    nmax = 30; CoefficientList[Series[Product[((1 + x^k)/(1 - x^k))^(2^(k - 1)), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

G.f.: Product_{k>=1} ((1 + x^k)/(1 - x^k))^A011782(k).
Euler transform of c(n) with g.f.: -x*(-2*x^2-x+2)/(-4*x^3+2*x^2+2*x-1). - Simon Plouffe, Jun 20 2018
a(n) ~ A247003 * 2^(n-1) * exp(2*sqrt(n) - 1/2 + c) / (sqrt(Pi)*n^(3/4)), where c = Sum_{k>=2} -(-1)^k / (k*(2^k-2)) = -0.207530918644117743551169251314627032... - Vaclav Kotesovec, Sep 15 2021

A319919 Expansion of Product_{k>=1} (1 + x^k)^(2^k-1).

Original entry on oeis.org

1, 1, 3, 10, 25, 70, 182, 476, 1220, 3122, 7883, 19794, 49340, 122237, 301114, 737923, 1799597, 4369204, 10563800, 25441377, 61048713, 145988775, 347981713, 826921992, 1959363778, 4629903905, 10911757432, 25652950459, 60165831361, 140792215037, 328750398275, 766041930160, 1781452975346
Offset: 0

Views

Author

Ilya Gutkovskiy, Oct 01 2018

Keywords

Comments

Convolution of A081362 and A102866.
Weigh transform of A000225.

Links

Crossrefs

Cf. A000225, A007070, A081362, A098407, A102866, A319918.

Programs

  • Maple
    a:=series(mul((1+x^k)^(2^k-1),k=1..100),x=0,33): seq(coeff(a,x,n),n=0..32); # Paolo P. Lava, Apr 02 2019
  • Mathematica
    nmax = 32; CoefficientList[Series[Product[(1 + x^k)^(2^k - 1), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 32; CoefficientList[Series[Exp[Sum[(-1)^(k + 1) x^k/(k (1 - x^k) (1 - 2 x^k)), {k, 1, nmax}]], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d (2^d - 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 32}]

Formula

G.f.: exp(Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - x^k)*(1 - 2*x^k))).
a(n) ~ c * exp(2*sqrt(n) - 1/2) * 2^(n-1) / (A079555 * sqrt(Pi) * n^(3/4)), where c = exp(Sum_{k>=2} (-1)^(k-1)/(k*(2^(k-1)-1))) = 0.6602994483152065685... - Vaclav Kotesovec, Sep 15 2021

A305209 a(n) = [x^n] exp(Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - n*x^k))).

Original entry on oeis.org

1, 1, 2, 12, 86, 885, 11234, 172711, 3112262, 64422126, 1506406702, 39279802969, 1130133725736, 35566642690293, 1215444767739120, 44823725114186355, 1774344335649148230, 75042087586212893216, 3377041177800135323864, 161125608740713509132809, 8124438293071792011560256
Offset: 0

Views

Author

Ilya Gutkovskiy, May 27 2018

Keywords

Links

Crossrefs

Cf. A098407, A292805, A305207.

Programs

  • Mathematica
    Table[SeriesCoefficient[Exp[Sum[(-1)^(k + 1) x^k/(k (1 - n x^k)), {k, 1, n}]], {x, 0, n}], {n, 0, 20}]
Table[SeriesCoefficient[Product[(1 + x^k)^(n^(k - 1)), {k, 1, n}], {x, 0, n}], {n, 0, 20}]

Formula

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

A305652 Expansion of Product_{k>=1} (1 + x^k)^(2^(k-1)-1).

Original entry on oeis.org

1, 0, 1, 3, 7, 18, 41, 99, 227, 538, 1236, 2872, 6597, 15166, 34669, 79150, 180011, 408616, 925015, 2089607, 4709937, 10595275, 23788174, 53312366, 119271967, 266399612, 594077742, 1322815256, 2941225084, 6530659320, 14481362803, 32070677496, 70937233268, 156721128440
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 07 2018

Keywords

Comments

Weigh transform of A000225, shifted right one place.
Convolution of the sequences A081362 and A098407.

Links

Crossrefs

Cf. A000225, A001906, A011782, A081362, A098407, A102866, A110045.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(2^(i-1)-1, j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..40);  # Alois P. Heinz, Jun 07 2018
  • Mathematica
    nmax = 33; CoefficientList[Series[Product[(1 + x^k)^(2^(k-1)-1), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 33; CoefficientList[Series[Exp[Sum[(-1)^(k + 1) x^(2 k)/(k (1 - x^k) (1 - 2 x^k)), {k, 1, nmax}]], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d (2^(d - 1) - 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 33}]

Formula

G.f.: Product_{k>=1} (1 + x^k)^A000225(k-1).
G.f.: Product_{k>=1} (1 + x^k)^(A011782(k)-1).
G.f.: exp(Sum_{k>=1} (-1)^(k+1)*x^(2*k)/(k*(1 - x^k)*(1 - 2*x^k))).
a(n) ~ 2^n * exp(sqrt(2*n) - 5/4 + c) / (sqrt(2*Pi) * 2^(3/4) * n^(3/4)), where c = Sum_{k>=2} -(-1)^k / (k*(2^k-1)*(2^k-2)) = -0.07640757130267274170429705262846... - Vaclav Kotesovec, Jun 08 2018
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