A216221 Triangular array read by rows. T(n,k) is the number of partitions of n (using 1 type of part 1, 2 types of part 2, ..., i types of part i, ...) that have exactly k distinct parts.

1, 3, 4, 2, 7, 6, 6, 17, 1, 12, 29, 7, 8, 55, 23, 15, 84, 58, 3, 13, 122, 134, 13, 18, 181, 249, 52, 12, 240, 464, 140, 3, 28, 321, 765, 348, 17, 14, 407, 1249, 746, 69, 24, 546, 1875, 1501, 220, 1, 24, 628, 2835, 2793, 586, 13, 31, 828, 4024, 4927, 1431, 56, 18, 940, 5707, 8331, 3123, 215, 39, 1211, 7741, 13520, 6436, 650, 4

Offset

1,2

Comments

Row sums = A000219.

Links

Table of n, a(n) for n=1..73.

P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; page 171

Formula

O.g.f.: Product_{i>=1} (1 + y*x^i/(1-x^i))^i.

Example

1,
3,
4,  2,
7,  6,
6,  17,  1,
12, 29,  7,
8,  55,  23,
15, 84,  58,   3,
13, 122, 134,  13,
18, 181, 249,  52,
12, 240, 464,  140,  3,
28, 321, 765,  348,  17,
14, 407, 1249, 746,  69,
24, 546, 1875, 1501, 220, 1
24, 628, 2835, 2793, 586, 13
T(4,2) = 6 because we have: 3+1, 3'+1, 3''+1, 2+2', 2+1+1, 2'+1+1.

Mathematica

nn=15; f[list_]:=Select[list, #>0&]; Map[f, Drop[CoefficientList[Series[ Product[(1+y x^i/(1-x^i))^i, {i, 1, nn}], {x, 0, nn}], {x, y}], 1]]//Grid

Crossrefs

Sequence in context: A205769 A166108 A255768 * A296431 A045901 A098003

Adjacent sequences: A216218 A216219 A216220 * A216222 A216223 A216224

Keyword

nonn,tabf

Author

Geoffrey Critzer, Mar 13 2013

Status

approved