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A209406
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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.
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8
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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
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OFFSET
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1,1
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COMMENTS
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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.
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LINKS
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Alois P. Heinz, Rows n = 1..141, flattened
Index entries for triangles generated by the Multiset Transformation
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FORMULA
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O.g.f.: Product_{i>=1} 1/(1-y*x^i)^(2^i).
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EXAMPLE
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: 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;
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MAPLE
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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
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MATHEMATICA
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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
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CROSSREFS
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Cf. A034899, A208741, A290222, A292506.
Sequence in context: A067179 A318993 A188843 * A188706 A304408 A048767
Adjacent sequences: A209403 A209404 A209405 * A209407 A209408 A209409
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KEYWORD
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nonn,tabl
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AUTHOR
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Geoffrey Critzer, Mar 08 2012
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STATUS
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approved
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