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 1-3 of 3 results.

A055922 Number of partitions of n in which each part occurs an odd number (or zero) times.

Original entry on oeis.org

1, 1, 1, 3, 2, 5, 6, 9, 9, 16, 20, 25, 32, 40, 54, 69, 84, 101, 136, 156, 202, 244, 306, 357, 448, 527, 652, 773, 944, 1103, 1346, 1574, 1885, 2228, 2640, 3106, 3684, 4302, 5052, 5931, 6924, 8079, 9416, 10958, 12718, 14824, 17078, 19820, 22860, 26433
Offset: 0

Views

Author

Christian G. Bower, Jun 23 2000

Keywords

Examples

			There exist 11 partitions of 6. For six of these partitions, each part occurs an odd number times, they are 6 = 5 + 1 = 4 + 2 = 3 + 2 + 1 = 3 + 1+1+1 = 2+2+2, hence a(6) = 6. The five other partitions are 4 + 1+1 = 3+3 = 2+2 + 1+1 = 2 + 1+1+1+1 = 1+1+1+1+1+1.

Links

Crossrefs

Cf. A000041, A007690, A055923, A249389.
Column k=0 of A264399.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(`if`(irem(j, 2)=0, 0, b(n-i*j, i-1)), j=1..n/i)
       +b(n, i-1)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..60);  # Alois P. Heinz, May 31 2014
  • Mathematica
    b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, Sum[If[Mod[j, 2] == 0, 0, b[n-i*j, i-1]], {j, 1, n/i}] + b[n, i-1]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Feb 24 2015, after Alois P. Heinz *)
  • PARI
    { my(n=60); Vec(prod(k=1, n, 1 + sum(r=0, n\(2*k), x^(k*(2*r+1))) + O(x*x^n))) } \\ Andrew Howroyd, Dec 22 2017

Formula

EULER transform of b where b has g.f. Sum {k>0} c(k)*x^k/(1-x^k) where c is inverse EULER transform of characteristic function of odd numbers.
G.f.: Product_{i>0} (1+x^i-x^(2*i))/(1-x^(2*i)). - Vladeta Jovovic, Feb 03 2004
Asymptotics (Auluck, Singwi, Agarwala, 1950): a(n) ~ B/(2*Pi*n) * exp(2*B*sqrt(n)), where B = sqrt(Pi^2/12 + 2*log(phi)^2), where phi is the golden ratio. - Vaclav Kotesovec, Oct 27 2014

A264398 Triangle read by rows: T(n,k) is the number of partitions of n having k parts with odd multiplicities.

Original entry on oeis.org

1, 0, 1, 1, 1, 0, 2, 1, 2, 2, 1, 0, 4, 3, 3, 4, 3, 1, 0, 7, 7, 1, 5, 7, 7, 3, 0, 12, 14, 4, 7, 12, 14, 8, 1, 0, 19, 26, 10, 1, 11, 19, 26, 18, 3, 0, 30, 45, 22, 4, 15, 30, 45, 36, 9, 0, 45, 75, 44, 11, 1, 22, 45, 75, 67, 21, 1, 0, 67, 120, 81, 26, 3, 30, 67, 120, 119, 45, 4
Offset: 0

Views

Author

Emeric Deutsch, Nov 21 2015

Keywords

Comments

T(n,0) = A035363(n).
Row sums give A000041. - Omar E. Pol, Nov 21 2015

Examples

			T(6,1) = 4 because we have [6*], [4*,1,1],[2*,2,2], and [2*,1,1,1,1] (parts with odd multiplicities are marked).
Triangle starts:
  1;
  0, 1;
  1, 1;
  0, 2, 1;
  2, 2, 1;
  0, 4, 3;
  3, 4, 3, 1;
  ...

Links

Crossrefs

Cf. A000041, A035363, A209423, A264399, A264400.

Programs

  • Maple
    g := product((1+t*x^j)/(1-x^(2*j)), j = 1 .. 100): gser := simplify(series(g, x = 0, 30)): for n from 0 to 28 do P[n] := sort(coeff(gser, x, n)) end do: for n from 0 to 23 do seq(coeff(P[n], t, j), j = 0 .. degree(P[n])) end do; # yields sequence in triangular form
# second Maple program:
b:= proc(n,i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(expand(`if`(j::odd, x, 1)*b(n-i*j, i-1)), j=0..n/i)))
    end:
T:= n-> (p-> seq(coeff(p,x,i), i=0..degree(p)))(b(n$2)):
seq(T(n), n=0..20);  # Alois P. Heinz, Nov 25 2015
  • Mathematica
    b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, Sum[Expand[If[OddQ[j], x, 1]* b[n-i*j, i-1]], {j, 0, n/i}]]]; T[n_] := Function[p, Table[Coefficient[ p, x, i], {i, 0, Exponent[p, x]}]][b[n, n]]; Table[T[n], {n, 0, 20}] // Flatten (* Jean-François Alcover, Feb 18 2016, after Alois P. Heinz *)
  • PARI
    T(n) = { Vec(prod(k=1, n, (1 + y*x^k)/(1 - x^(2*k)) + O(x*x^n))) }
{ my(t=T(10)); for(n=1, #t, print(Vecrev(t[n]))); } \\ Andrew Howroyd, Dec 22 2017

Formula

G.f.: G(t,x) = Product_{j>=1} (1 + tx^j)/(1 - x^(2j)).
Sum_{k>0} k * T(n,k) = A209423(n). - Alois P. Heinz, Aug 05 2020
G.f.: A(x,y)*B(x^2) where A(x),B(x) are the o.g.f.'s for A008289 and A000041. (See Flajolet, Sedgewick link.) - Geoffrey Critzer, Aug 07 2022

A264400 Number of parts of even multiplicities in all the partitions of n.

Original entry on oeis.org

0, 0, 1, 0, 3, 2, 6, 6, 15, 15, 29, 34, 58, 70, 109, 132, 199, 246, 348, 435, 601, 746, 1005, 1252, 1653, 2053, 2666, 3298, 4231, 5219, 6608, 8124, 10198, 12476, 15525, 18927, 23374, 28387, 34823, 42122, 51376, 61922, 75098, 90200, 108874, 130298, 156564, 186777, 223490, 265779, 316799
Offset: 0

Views

Author

Emeric Deutsch, Nov 21 2015

Keywords

Comments

a(n) = Sum_{k>=0} k*A264399(n,k).

Examples

			a(6) = 6 because we have [6], [5,1], [4,2], [4,1*,1], [3*,3], [3,2,1], [3,1,1,1], [2,2,2], [2*,2,1*,1], [2,1*,1,1,1], and [1*,1,1,1,1,1] (the 6 parts with even multiplicities are marked).

Links

Crossrefs

Cf. A264399, A325939.

Programs

  • Maple
    g := (sum(x^(2*j)/(1+x^j), j = 1 .. 100))/(product(1-x^j, j = 1 .. 100)): gser := series(g, x = 0, 70): seq(coeff(gser, x, n), n = 0 .. 60);
  • Mathematica
    Needs["Combinatorica`"]; Table[Count[Last /@ Flatten[Tally /@ Combinatorica`Partitions@ n, 1], k_ /; EvenQ@ k], {n, 0, 50}] (* Michael De Vlieger, Nov 21 2015 *)
Table[Sum[(1 - 2*DivisorSigma[0, 2*k] + 3*DivisorSigma[0, k]) * PartitionsP[n-k], {k, 1, n}], {n, 0, 50}] (* Vaclav Kotesovec, Jun 14 2025 *)
  • PARI
    { my(n=50); Vec(sum(k=1, n, x^(2*k)/(1+x^k) + O(x*x^n)) / prod(k=1, n, 1-x^k + O(x*x^n)), -(n+1)) } \\ Andrew Howroyd, Dec 22 2017

Formula

G.f.: g(x) = (Sum_{j>=1} (x^(2j)/(1+x^j))) / Product_{k>=1} (1-x^k).
Showing 1-3 of 3 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.