

A126442


Triangular array t read by rows: t(0,k) is p(k), the number of partitions of the kmultiset {0,0,...,0} with k zeros. For 0 <= n < k, t(n, k) is the number of partitions of the kmultiset {0, 0, ..., 0, 1, 2, 3, ..., kn} with n zeros.


9



1, 2, 2, 3, 4, 5, 5, 7, 11, 15, 7, 12, 21, 36, 52, 11, 19, 38, 74, 135, 203, 15, 30, 64, 141, 296, 566, 877, 22, 45, 105, 250, 592, 1315, 2610, 4140, 30, 67, 165, 426, 1098, 2752, 6393, 13082, 21147, 42, 97, 254, 696, 1940, 5317, 13960, 33645, 70631, 115975
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OFFSET

1,2


COMMENTS

First in a series of triangular arrays which comprise subsequences of A096443(n).
The second array begins 9 16 26 29 52 92 47 98 198 371 and when the arrays are aligned as illustrated in triangle A126441 with p(n) values they sum to A035310 which counts unordered multisets.
Let t(n, k) be the number of ways to partition the kmultiset {0,0,...,0,1,2,3,4,...,kn} with n zeros, 0 <= n < k. Then t(n, k) = sum_i = 0..k j = 0..n S(n, j) C(i, j) p(k  n  i), where S(n, j) are Stirling numbers of the second kind, C(i, j) are the number of compositions of i distinct objects into j parts, and p is the integer partition function.
To see this, partition [n] into j blocks; there are S(n, j) partitions. For such a partition x and for each i, there are C(i, j) ways to distribute i zeros into x, because the blocks of x are all distinct. There are p(kni) ways to partition the remaining kni zeros. Multiplying and summing gives the result.  George Beck, Jan 10 2011
Values are also part of A096443, A129306 and A249620. Columns are also columns of the last one of these irregular triangles. See "Partitions_of_multisets" link.  Tilman Piesk, Nov 09 2014


LINKS

Table of n, a(n) for n=1..55.
Tilman Piesk, Partitions of multisets (Wikiversity)


EXAMPLE

This first array includes only the hook cases. A096443(9,14,16) correspond to partitions [2,2], [3,2] and [2,2,1] so these values do not appear in A126442.
The array begins:
1
2 2
3 4 5
5 7 11 15
7 12 21 36 52


MATHEMATICA

(* The triangle is flattened to a sequence. *)
t[n_, k_] := Sum[StirlingS2[n, j] * Binomial[1 + i + j, i] * PartitionsP[k  n  i], {j, 0, n}, {i, 0, k  n}]; Table[ t[n, k], {k, 10}, {n, 0, k  1}] // Flatten (* George Beck, Jan 10 2011 *)


CROSSREFS

Cf. A000041, A000070, A082775, A093802, A000291, A002763, A000412, A054225, A035310, A000110, A035098.
Sequence in context: A015743 A015755 A096443 * A129306 A322077 A114094
Adjacent sequences: A126439 A126440 A126441 * A126443 A126444 A126445


KEYWORD

nonn,tabl


AUTHOR

Alford Arnold, Jan 28 2007


EXTENSIONS

Definition clarified by George Beck, Jan 11 2011


STATUS

approved



