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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A210485 Number T(n,k) of parts in all partitions of n in which no part occurs more than k times; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

0, 0, 1, 0, 1, 3, 0, 3, 3, 6, 0, 3, 8, 8, 12, 0, 5, 11, 15, 15, 20, 0, 8, 17, 24, 29, 29, 35, 0, 10, 23, 36, 41, 47, 47, 54, 0, 13, 36, 50, 65, 71, 78, 78, 86, 0, 18, 48, 75, 91, 104, 111, 119, 119, 128, 0, 25, 69, 102, 132, 150, 165, 173, 182, 182, 192
Offset: 0

Views

Author

Alois P. Heinz, Jan 23 2013

Keywords

Comments

T(n,k) is defined for n,k >= 0. The triangle contains terms with k <= n. T(n,k) = T(n,n) = A006128(n) for k >= n.
For fixed k > 0, T(n,k) ~ 3^(1/4) * log(k+1) * exp(Pi*sqrt(2*k*n/(3*(k+1)))) / (Pi * (8*k*(k+1)*n)^(1/4)). - Vaclav Kotesovec, Oct 18 2018

Examples

			T(6,2) = 17: [6], [5,1], [4,2], [3,3], [4,1,1], [3,2,1], [2,2,1,1].
Triangle T(n,k) begins:
  0;
  0,  1;
  0,  1,  3;
  0,  3,  3,  6;
  0,  3,  8,  8, 12;
  0,  5, 11, 15, 15, 20;
  0,  8, 17, 24, 29, 29, 35;
  0, 10, 23, 36, 41, 47, 47, 54;
  0, 13, 36, 50, 65, 71, 78, 78, 86;
  ...

Links

Crossrefs

Columns k=0-10 give: A000004, A015723, A185350, A117148, A320607, A320608, A320609, A320610, A320611, A320612, A320613.
Main diagonal gives A006128.
T(2n,n) gives A364245.
Cf. A213177, A286653.

Programs

  • Maple
    b:= proc(n, i, k) option remember; `if`(n=0, [1, 0], `if`(i<1, [0, 0],
      add((l->[l[1], l[2]+l[1]*j])(b(n-i*j, i-1, k)), j=0..min(n/i, k))))
    end:
T:= (n, k)-> b(n, n, k)[2]:
seq(seq(T(n, k), k=0..n), n=0..12);
  • Mathematica
    b[n_, i_, k_] := b[n, i, k] = If[n == 0, {1, 0}, If[i < 1, {0, 0}, Sum[b[n-i*j, i-1, k] /. l_List :> {l[[1]], l[[2]] + l[[1]]*j}, {j, 0, Min[n/i, k]}]]]; T[n_, k_] := b[n, n, k][[2]]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Dec 27 2013, translated from Maple *)

Formula

T(n,k) = Sum_{i=0..k} A213177(n,i).

A117147 Triangle read by rows: T(n,k) is the number of partitions of n with k parts in which no part occurs more than 3 times (n>=1, k>=1).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 3, 3, 2, 1, 3, 4, 3, 1, 1, 4, 5, 4, 2, 1, 4, 7, 6, 3, 1, 1, 5, 8, 9, 5, 1, 1, 5, 10, 11, 8, 3, 1, 6, 12, 14, 11, 5, 1, 1, 6, 14, 18, 15, 8, 2, 1, 7, 16, 23, 20, 11, 4, 1, 7, 19, 27, 27, 17, 6, 1, 1, 8, 21, 33, 34, 23, 10, 2, 1, 8, 24, 39, 43, 32, 15, 4, 1, 9
Offset: 1

Views

Author

Emeric Deutsch, Mar 07 2006

Keywords

Comments

Row n has floor(sqrt(6n+6)-3/2) terms. Row sums yield A001935. Sum(k*T(n,k),k>=0) = A117148(n).

Examples

			T(7,3) = 4 because we have [5,1,1], [4,2,1], [3,3,1] and [3,2,2].
Triangle starts:
1;
1, 1;
1, 1, 1;
1, 2, 1;
1, 2, 2, 1;
1, 3, 3, 2;
1, 3, 4, 3, 1;

Links

Crossrefs

Cf. A001935, A008289, A117148, A209318.

Programs

  • Maple
    g:=-1+product(1+t*x^j+t^2*x^(2*j)+t^3*x^(3*j),j=1..35): gser:=simplify(series(g,x=0,23)): for n from 1 to 18 do P[n]:=sort(coeff(gser,x^n)) od: for n from 1 to 18 do seq(coeff(P[n],t^j),j=1..floor(sqrt(6*n+6)-3/2)) od; # yields sequence in triangular form
# second Maple program
b:= proc(n, i) option remember; local j; if n=0 then 1
      elif i<1 then 0 else []; for j from 0 to min(3, n/i) do
      zip((x, y)->x+y, %, [0$j, b(n-i*j, i-1)], 0) od; %[] fi
    end:
T:= n-> subsop(1=NULL, [b(n, n)])[]:
seq(T(n), n=1..20);  # Alois P. Heinz, Jan 08 2013
  • Mathematica
    max = 18; g = -1+Product[1+t*x^j+t^2*x^(2j)+t^3*x^(3j), {j, 1, max}]; t[n_, k_] := SeriesCoefficient[g, {x, 0, n}, {t, 0, k}]; Table[DeleteCases[Table[t[n, k], {k, 1, n}], 0], {n, 1, max}] // Flatten (* Jean-François Alcover, Jan 08 2014 *)

Formula

G.f.: G(t,x) = -1+product(1+tx^j+t^2*x^(2j)+t^3*x^(3j), j=1..infinity).

A185350 Number of parts in all partitions of n in which no part occurs more than twice.

Original entry on oeis.org

0, 1, 3, 3, 8, 11, 17, 23, 36, 48, 69, 88, 125, 157, 212, 271, 356, 445, 574, 711, 906, 1118, 1400, 1711, 2125, 2583, 3171, 3828, 4666, 5604, 6777, 8095, 9730, 11567, 13815, 16357, 19429, 22910, 27077, 31801, 37432, 43802, 51338, 59871, 69914, 81273, 94562
Offset: 0

Views

Author

Alois P. Heinz, Jan 21 2013

Keywords

Examples

			a(6) = 17: [6], [5,1], [4,2], [3,3], [4,1,1], [3,2,1], [2,2,1,1].
a(7) = 23: [7], [6,1], [5,2], [4,3], [5,1,1], [4,2,1], [3,3,1], [3,2,2], [3,2,1,1].

Links

Crossrefs

Column k=2 of A210485.
Cf. A015723, A117148, A209318.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, [0, 0],
      add((l->[l[1], l[2]+l[1]*j])(b(n-i*j, i-1)), j=0..min(n/i, 2))))
    end:
a:= n-> b(n, n)[2]:
seq(a(n), n=0..50);
  • Mathematica
    b[n_, i_, k_] := b[n, i, k] = If[n==0, {1, 0}, If[i<1, {0, 0}, Sum[b[n - i j, i - 1, k] /. l_List :> {l[[1]], l[[2]] + l[[1]] j}, {j, 0, Min[n/i, k]} ] ] ];
a[n_] := b[n, n, 2][[2]];
a /@ Range[0, 50] (* Jean-François Alcover, Dec 10 2020, after Alois P. Heinz *)
Table[Length[Flatten[Select[IntegerPartitions[n],Max[Length/@Split[#]]<3&]]],{n,0,50}] (* Harvey P. Dale, Jul 04 2023 *)

Formula

a(n) = Sum_{k>=0} k*A209318(n,k).
a(n) ~ log(3) * exp(2*Pi*sqrt(n)/3) / (2*Pi*n^(1/4)). - Vaclav Kotesovec, May 26 2018
Showing 1-3 of 3 results.

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