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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A218074 Expansion of Sum_{n>=1} ((n-1) * q^(n*(n+1)/2) / Product_{k=1..n} (1 - q^k)).

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 4, 5, 7, 10, 15, 18, 25, 31, 41, 53, 66, 81, 103, 125, 154, 190, 229, 276, 333, 399, 475, 568, 673, 794, 938, 1102, 1289, 1512, 1760, 2050, 2384, 2760, 3190, 3687, 4246, 4882, 5609, 6427, 7354, 8412, 9592, 10927, 12439, 14130, 16033, 18177, 20573, 23256, 26271
Offset: 0

Views

Author

Joerg Arndt, Oct 20 2012

Keywords

Comments

Number of up-steps (== number of parts - 1) in all partitions of n into distinct parts (represented as increasing lists), see example. - Joerg Arndt, Sep 03 2014

Examples

			a(8) = 7 because in the 6 partitions of 8 into distinct parts
  1:  [ 1 2 5 ]
  2:  [ 1 3 4 ]
  3:  [ 1 7 ]
  4:  [ 2 6 ]
  5:  [ 3 5 ]
  6:  [ 8 ]
there are 2+2+1+1+1+0 = 7 up-steps. - _Joerg Arndt_, Sep 03 2014

Links

Crossrefs

Cf. A015723, Sum_{n>=0} (n * q^(n*(n+1)/2) / Product_{k=1..n} (1 - q^k)).
Cf. A032020, Sum_{n>=0} (n! * q^(n*(n+1)/2) / Product_{k=1..n} (1 - q^k)).
Cf. A032153, Sum_{n>=1} ((n-1)! * q^(n*(n+1)/2) / Product_{k=1..n} (1 - q^k)).
Cf. A072576, Sum_{n>=0} ((n+1)! * q^(n*(n+1)/2) / Product_{k=1..n} (1 - q^k)).
Cf. A058884 (up-steps in all partitions).

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, 0,
      b(n, i-1)+`if`(i>n, 0, (p->p+[0, p[1]])(b(n-i, i-1)))))
    end:
a:= n-> `if`(n=0, 0, (p-> p[2]-p[1])(b(n$2))):
seq(a(n), n=0..80);  # Alois P. Heinz, Sep 03 2014
  • Mathematica
    max=80; s=Sum[(n-1)*q^(n*(n+1)/2)/QPochhammer[q, q, n], {n, Sqrt[max+1]}]+ O[q]^max; CoefficientList[s, q] (* Jean-François Alcover, Jan 17 2016 *)
  • PARI
    N=66;  q='q+O('q^N);
gf=sum(n=1,N, (n-1)*q^(n*(n+1)/2) / prod(k=1,n, 1-q^k ) );
v=Vec(gf+'a0);  v[1]-='a0;  v  /* include initial zeros */

Formula

a(n) = A015723(n) - A000009(n) for n>0. - Alois P. Heinz, Sep 03 2014

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.

A336896 Sum of the leftmost parts in all compositions of n into distinct parts.

Original entry on oeis.org

0, 1, 2, 6, 8, 15, 30, 42, 64, 99, 190, 242, 384, 533, 798, 1380, 1824, 2635, 3762, 5320, 7280, 12327, 15554, 22632, 30720, 43425, 57538, 80730, 122920, 159239, 220830, 299150, 406656, 542883, 733278, 962710, 1443600, 1820437, 2496638, 3280992, 4451120
Offset: 0

Views

Author

Alois P. Heinz, Aug 07 2020

Keywords

Comments

Also sum of the rightmost parts in all compositions of n into distinct parts.

Examples

			a(6) = 30 = 1 + 1 + 2 + 2 + 3 + 3 + 2 + 4 + 1 + 5 + 6: (1)23, (1)32, (2)13, (2)31, (3)12, (3)21, (2)4, (4)2, (1)5, (5)1, (6).

Links

Crossrefs

Cf. A000225 (the same for all compositions), A008289, A032020, A032153.

Programs

  • Maple
    b:= proc(n, i, p) option remember; `if`(i*(i+1)/2 `if`(n=0, 0, n*b(n$2, -1)):
seq(a(n), n=0..50);
  • Mathematica
    b[n_, i_, p_] := b[n, i, p] = If[i*(i + 1)/2 < n, 0,
     If[n == 0, p!, b[n, i - 1, p] + b[n - i, Min[n - i, i - 1], p + 1]]];
a[n_] := If[n == 0, 0, n*b[n, n, -1]];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Apr 13 2022, after Alois P. Heinz *)

Formula

a(n) = n * Sum_{k=1..floor((sqrt(8*n+1)-1)/2)} (k-1)! * A008289(n,k).
a(n) = n * A032153(n).
Previous Showing 11-16 of 16 results.

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