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

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

0,10

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.

Links

Alois P. Heinz, Rows n = 0..140, flattened

Eric Weisstein's World of Mathematics, Partition Function Q.

Formula

T(n,k) = b(0,n,k), where b(m,n,k) = 1+sum(b(i,j,k): m<i<j<k and i+j=n).

T(n,k) = coefficient of x^n in product_{i=1..k} (1+x^i). - Vladeta Jovovic, Aug 07 2003

Example

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; ...

Maple

with(numtheory):
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 *)

Crossrefs

Cf. A000009, A079122, A035294, A079124, A079125.

Differs from A026840 in having extra zeros at the ends of the rows.

Sequence in context: A132406 A197881 A332712 * A339086 A186336 A025891

Adjacent sequences: A079123 A079124 A079125 * A079127 A079128 A079129

Keyword

nonn,tabl

Author

Reinhard Zumkeller, Dec 27 2002

Status

approved