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A209406
Triangular array read by rows: T(n,k) is the number of multisets of exactly k nonempty binary words with a total of n letters.
10
2, 4, 3, 8, 8, 4, 16, 26, 12, 5, 32, 64, 44, 16, 6, 64, 164, 132, 62, 20, 7, 128, 384, 376, 200, 80, 24, 8, 256, 904, 1008, 623, 268, 98, 28, 9, 512, 2048, 2632, 1792, 870, 336, 116, 32, 10, 1024, 4624, 6624, 5040, 2632, 1117, 404, 134, 36, 11
OFFSET
1,1
COMMENTS
Equivalently, T(n,k) is the number of partitions of the integer n with two types of 1's, four types of 2's, ..., 2^i types of i's...; having exactly k summands (of any type).
Row sums = A034899.
FORMULA
O.g.f.: Product_{i>=1} 1/(1-y*x^i)^(2^i).
EXAMPLE
Triangle T(n,k) begins:
2;
4, 3;
8, 8, 4;
16, 26, 12, 5;
32, 64, 44, 16, 6;
64, 164, 132, 62, 20, 7;
128, 384, 376, 200, 80, 24, 8;
256, 904, 1008, 623, 268, 98, 28, 9;
512, 2048, 2632, 1792, 870, 336, 116, 32, 10;
...
MAPLE
b:= proc(n, i, p) option remember; `if`(p>n, 0, `if`(n=0, 1,
`if`(min(i, p)<1, 0, add(b(n-i*j, i-1, p-j)*
binomial(2^i+j-1, j), j=0..min(n/i, p)))))
end:
T:= (n, k)-> b(n$2, k):
seq(seq(T(n, k), k=1..n), n=1..14); # Alois P. Heinz, Apr 13 2017
MATHEMATICA
nn = 10; p[x_, y_] := Product[1/(1 - y x^i)^(2^i), {i, 1, nn}]; f[list_] := Select[lst, # > 0 &]; Map[f, Drop[CoefficientList[Series[p[x, y], {x, 0, nn}], {x, y}], 1]] // Flatten
CROSSREFS
T(2n,n) gives A359962.
Sequence in context: A318993 A355482 A188843 * A188706 A304408 A048767
KEYWORD
nonn,tabl
AUTHOR
Geoffrey Critzer, Mar 08 2012
STATUS
approved