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A208741
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Triangular array read by rows. T(n,k) is the number of sets of exactly k distinct binary words with a total of n letters.
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9
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2, 4, 1, 8, 8, 16, 22, 4, 32, 64, 20, 64, 156, 84, 6, 128, 384, 264, 40, 256, 888, 784, 189, 4, 512, 2048, 2152, 704, 50, 1024, 4592, 5664, 2384, 272, 1, 2048, 10240, 14368, 7328, 1232, 32, 4096, 22496, 35568, 21382, 4704, 248
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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 integer partitions of n into distinct parts with two types of 1's, four types of 2's, ... , 2^i types of i's,...; where k is the number of summands (of any type).
Row lengths increase by 1 at n=A061168(offset).
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LINKS
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FORMULA
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O.g.f.: Product_{i>=1} (1 + y*x^i)^(2^i).
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EXAMPLE
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T(3,2) = 8 because we have: {a,aa}, {a,ab}, {a,ba}, {a,bb}, {b,aa}, {b,ab}, {b,ba}, {b,bb}; 2 word languages with total length 3.
Triangle T(n,k) begins:
2;
4, 1;
8, 8;
16, 22, 4;
32, 64, 20;
64, 156, 84, 6;
...
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MAPLE
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h:= proc(n, i) option remember; expand(`if`(n=0, 1, `if`(i<1, 0,
add(h(n-i*j, i-1)*binomial(2^i, j)*x^j, j=0..n/i))))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(h(n$2)):
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MATHEMATICA
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nn=12; p=Product[(1+y x^i)^(2^i), {i, 1, nn}]; f[list_] := Select[list, #>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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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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