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

0,4

Comments

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

Links

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

Formula

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

Example

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;

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 *)

Crossrefs

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

Main diagonal gives: A000041.

Sequence in context: A029269 A352166 A272187 * A088004 A070548 A209628

Adjacent sequences: A194618 A194619 A194620 * A194622 A194623 A194624

Keyword

nonn,tabl

Author

Alois P. Heinz, Aug 30 2011

Status

approved