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Number T(n,k) of multisets of exactly k nonempty binary words with a total of n letters such that no word has a majority of 0's; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
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%I #25 Aug 12 2020 18:53:28

%S 1,0,1,0,3,1,0,4,3,1,0,11,10,3,1,0,16,23,10,3,1,0,42,59,33,10,3,1,0,

%T 64,134,83,33,10,3,1,0,163,320,230,98,33,10,3,1,0,256,699,568,270,98,

%U 33,10,3,1,0,638,1599,1451,738,291,98,33,10,3,1,0,1024,3434,3439,1935,798,291,98,33,10,3,1

%N Number T(n,k) of multisets of exactly k nonempty binary words with a total of n letters such that no word has a majority of 0's; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

%H Alois P. Heinz, <a href="/A292506/b292506.txt">Rows n = 0..140, flattened</a>

%H <a href="/index/Mu#multiplicative_completely">Index entries for triangles generated by the Multiset Transformation</a>

%F G.f.: Product_{j>=1} 1/(1-y*x^j)^A027306(j).

%e T(4,2) = 10: {1,011}, {1,101}, {1,110}, {1,111}, {01,01}, {01,10}, {01,11}, {10,10}, {10,11}, {11,11}.

%e Triangle T(n,k) begins:

%e 1;

%e 0, 1;

%e 0, 3, 1;

%e 0, 4, 3, 1;

%e 0, 11, 10, 3, 1;

%e 0, 16, 23, 10, 3, 1;

%e 0, 42, 59, 33, 10, 3, 1;

%e 0, 64, 134, 83, 33, 10, 3, 1;

%e 0, 163, 320, 230, 98, 33, 10, 3, 1;

%e 0, 256, 699, 568, 270, 98, 33, 10, 3, 1;

%e 0, 638, 1599, 1451, 738, 291, 98, 33, 10, 3, 1;

%e ...

%p g:= n-> 2^(n-1)+`if`(n::odd, 0, binomial(n, n/2)/2):

%p b:= proc(n, i) option remember; expand(`if`(n=0 or i=1, x^n,

%p add(binomial(g(i)+j-1, j)*b(n-i*j, i-1)*x^j, j=0..n/i)))

%p end:

%p T:= n-> (p-> seq(coeff(p,x,i), i=0..n))(b(n$2)):

%p seq(T(n), n=0..12);

%t g[n_] := 2^(n-1) + If[OddQ[n], 0, Binomial[n, n/2]/2];

%t b[n_, i_] := b[n, i] = Expand[If[n == 0 || i == 1, x^n, Sum[Binomial[g[i] + j - 1, j]*b[n - i*j, i - 1]*x^j, {j, 0, n/i}]]];

%t T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n}]][b[n, n]];

%t Table[T[n], {n, 0, 12}] // Flatten (* _Jean-François Alcover_, Jun 06 2018, from Maple *)

%Y Columns k=0-10 give: A000007, A027306 (for n>0), A316403, A316404, A316405, A316406, A316407, A316408, A316409, A316410, A316411.

%Y Row sums give A292548.

%Y T(2n,n) gives A292549.

%Y Cf. A209406, A226873, A290222.

%K nonn,tabl

%O 0,5

%A _Alois P. Heinz_, Sep 17 2017