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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. 0
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Row sums = A000219.
LINKS
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
KEYWORD
nonn,tabf
AUTHOR
Geoffrey Critzer, Mar 13 2013
STATUS
approved

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Last modified May 29 00:29 EDT 2024. Contains 372921 sequences. (Running on oeis4.)