A194621 Triangle T(n,k), n>=0, 0<=k<=n, read by rows: T(n,k) is the number of partitions of n in which any two parts differ by at most k.

1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 4, 5, 5, 5, 2, 5, 6, 7, 7, 7, 4, 6, 9, 10, 11, 11, 11, 2, 7, 10, 13, 14, 15, 15, 15, 4, 8, 14, 17, 20, 21, 22, 22, 22, 3, 9, 15, 22, 25, 28, 29, 30, 30, 30, 4, 10, 20, 27, 34, 37, 40, 41, 42, 42, 42, 2, 11, 21, 33, 41, 48, 51, 54, 55, 56, 56, 56

Offset: 0

Author

Alois P. Heinz, Aug 30 2011

Keywords

• nonn
• tabl

Comments

T(n,k) = A000041(n) for n >= 0 and k >= n.

Examples

			T(6,0) = 4: [6], [3,3], [2,2,2], [1,1,1,1,1,1].
T(6,1) = 6: [6], [3,3], [2,1,1,1,1], [2,2,1,1], [2,2,2], [1,1,1,1,1,1].
T(6,2) = 9: [6], [4,2], [3,1,1,1], [3,2,1], [3,3], [2,1,1,1,1], [2,2,1,1], [2,2,2], [1,1,1,1,1,1].
Triangle begins:
  1;
  1, 1;
  2, 2,  2;
  2, 3,  3,  3;
  3, 4,  5,  5,  5;
  2, 5,  6,  7,  7,  7;
  4, 6,  9, 10, 11, 11, 11;
  2, 7, 10, 13, 14, 15, 15, 15;
		

Links

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

Crossrefs

Columns k=0-10 give (for n>0): A000005, A000027, A117142, A117143, A218506, A218507, A218508, A218509, A218510, A218511, A218512.

Main diagonal gives: A000041.

Programs

Maple

b:= proc(n, i, k) option remember;
      if n<0 or k<0 then 0
    elif n=0 then 1
    elif i<1 then 0
    else b(n, i-1, k-1) +b(n-i, i, k)
      fi
    end:
T:= (n, k)-> `if`(n=0, 1, 0) +add(b(n-i, i, k), i=1..n):
seq(seq(T(n, k), k=0..n), n=0..20);

Mathematica

b[n_, i_, k_] := b[n, i, k] = If[n < 0 || k < 0, 0, If[n == 0, 1, If[i < 1, 0, b[n, i-1, k-1] + b[n-i, i, k]]]]; t[n_, k_] := If[n == 0, 1, 0] + Sum[b[n-i, i, k], {i, 1, n}]; Table[Table[t[n, k], {k, 0, n}], {n, 0, 20}] // Flatten (* Jean-François Alcover, Dec 09 2013, translated from Maple *)

Formula

G.f. of column k: 1 + Sum_{j>0} x^j / Product_{i=0..k} (1-x^(i+j)).