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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.

A079126 Triangle T(n,k) of numbers of partitions of n into distinct positive integers <= k, 0<=k<=n.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 0, 1, 2, 0, 0, 0, 1, 2, 0, 0, 0, 1, 2, 3, 0, 0, 0, 1, 2, 3, 4, 0, 0, 0, 0, 2, 3, 4, 5, 0, 0, 0, 0, 1, 3, 4, 5, 6, 0, 0, 0, 0, 1, 3, 5, 6, 7, 8, 0, 0, 0, 0, 1, 3, 5, 7, 8, 9, 10, 0, 0, 0, 0, 0, 2, 5, 7, 9, 10, 11, 12, 0, 0, 0, 0, 0, 2, 5, 8, 10, 12, 13, 14, 15, 0, 0, 0, 0, 0, 1, 4, 8, 11, 13, 15, 16, 17, 18
Offset: 0

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Author

Reinhard Zumkeller, Dec 27 2002

Keywords

Comments

T(n,n) = A000009(n), right side of the triangle;
T(n,k)=0 for n>0 and k < A002024(n); T(prime(n),n) = A067953(n) for n>0.

Examples

			The seven partitions of n=5 are {5}, {4,1}, {3,2}, {3,1,1}, {2,2,1}, {2,1,1,1} and {1,1,1,1,1}. Only two of them ({4,1} and {3,2}) have distinct parts <= 4, so T(5,4) = 2.
Triangle T(n,k) begins:
1;
0, 1;
0, 0, 1;
0, 0, 1, 2;
0, 0, 0, 1 ,2;
0, 0, 0, 1, 2, 3;
0, 0, 0, 1, 2, 3, 4;
0, 0, 0, 0, 2, 3, 4, 5;
0, 0, 0, 0, 1, 3, 4, 5,  6;
0, 0, 0, 0, 1, 3, 5, 6,  7,  8;
0, 0, 0, 0, 1, 3, 5, 7,  8,  9, 10;
0, 0, 0, 0, 0, 2, 5, 7,  9, 10, 11, 12;
0, 0, 0, 0, 0, 2, 5, 8, 10, 12, 13, 14, 15; ...

Links

Crossrefs

Cf. A000009, A079122, A035294, A079124, A079125.
Differs from A026840 in having extra zeros at the ends of the rows.

Programs

  • Maple
    T:= proc(n, i) option remember; `if`(n=0, 1,
      `if`(i<1, 0, T(n, i-1)+`if`(i>n, 0, T(n-i, i-1))))
    end:
seq(seq(T(n,k), k=0..n), n=0..20);  # Alois P. Heinz, May 11 2015
  • Mathematica
    T[n_, i_] := T[n, i] = If[n==0, 1, If[i<1, 0, T[n, i-1] + If[i>n, 0, T[n-i, i-1]]]]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 20}] // Flatten (* Jean-François Alcover, Jun 30 2015, after Alois P. Heinz *)

Formula

T(n,k) = b(0,n,k), where b(m,n,k) = 1+sum(b(i,j,k): m
T(n,k) = coefficient of x^n in product_{i=1..k} (1+x^i). - Vladeta Jovovic, Aug 07 2003

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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