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A290222
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Multiset transform of A011782, powers of 2: 1, 2, 4, 8, 16, ...
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5
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1, 0, 1, 0, 2, 1, 0, 4, 2, 1, 0, 8, 7, 2, 1, 0, 16, 16, 7, 2, 1, 0, 32, 42, 20, 7, 2, 1, 0, 64, 96, 54, 20, 7, 2, 1, 0, 128, 228, 140, 59, 20, 7, 2, 1, 0, 256, 512, 360, 156, 59, 20, 7, 2, 1, 0, 512, 1160, 888, 422, 162, 59, 20, 7, 2, 1, 0, 1024, 2560, 2168, 1088, 442, 162, 59, 20, 7, 2, 1
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OFFSET
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0,5
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COMMENTS
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T(n,k) is the number of multisets of exactly k binary words with a total of n letters and each word beginning with 1. T(4,2) = 7: {1,100}, {1,101}, {1,110}, {1,111}, {10,10}, {10,11}, {11,11}. - Alois P. Heinz, Sep 18 2017
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LINKS
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FORMULA
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G.f.: 1 / Product_{j>=1} (1-y*x^j)^(2^(j-1)). - Alois P. Heinz, Sep 18 2017
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EXAMPLE
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The triangle starts:
1;
0 1;
0 2 1;
0 4 2 1;
0 8 7 2 1;
0 16 16 7 2 1;
0 32 42 20 7 2 1;
0 64 96 54 20 7 2 1;
0 128 228 140 59 20 7 2 1;
0 256 512 360 156 59 20 7 2 1;
0 512 1160 888 422 162 59 20 7 2 1;
0 1024 2560 2168 1088 442 162 59 20 7 2 1;
(...)
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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(binomial(2^(i-1)+j-1, j)*
b(n-i*j, i-1, p-j), j=0..min(n/i, p)))))
end:
T:= (n, k)-> b(n$2, k):
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MATHEMATICA
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b[n_, i_, p_] := b[n, i, p] = If[p > n, 0, If[n == 0, 1, If[Min[i, p] < 1, 0, Sum[Binomial[2^(i - 1) + j - 1, j] b[n - i j, i - 1, p - j], {j, 0, Min[n/i, p]}]]]];
T[n_, k_] := b[n, n, k];
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CROSSREFS
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The reverse of the n-th row converges to A034899.
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KEYWORD
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AUTHOR
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STATUS
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approved
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