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A292506 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.

%I A292506 #25 Aug 12 2020 18:53:28
%S A292506 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 A292506 64,134,83,33,10,3,1,0,163,320,230,98,33,10,3,1,0,256,699,568,270,98,
%U A292506 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 A292506 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 A292506 Alois P. Heinz, <a href="/A292506/b292506.txt">Rows n = 0..140, flattened</a>
%H A292506 <a href="/index/Mu#multiplicative_completely">Index entries for triangles generated by the Multiset Transformation</a>
%F A292506 G.f.: Product_{j>=1} 1/(1-y*x^j)^A027306(j).
%e A292506 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 A292506 Triangle T(n,k) begins:
%e A292506   1;
%e A292506   0,   1;
%e A292506   0,   3,    1;
%e A292506   0,   4,    3,    1;
%e A292506   0,  11,   10,    3,   1;
%e A292506   0,  16,   23,   10,   3,   1;
%e A292506   0,  42,   59,   33,  10,   3,  1;
%e A292506   0,  64,  134,   83,  33,  10,  3,  1;
%e A292506   0, 163,  320,  230,  98,  33, 10,  3,  1;
%e A292506   0, 256,  699,  568, 270,  98, 33, 10,  3, 1;
%e A292506   0, 638, 1599, 1451, 738, 291, 98, 33, 10, 3, 1;
%e A292506   ...
%p A292506 g:= n-> 2^(n-1)+`if`(n::odd, 0, binomial(n, n/2)/2):
%p A292506 b:= proc(n, i) option remember; expand(`if`(n=0 or i=1, x^n,
%p A292506       add(binomial(g(i)+j-1, j)*b(n-i*j, i-1)*x^j, j=0..n/i)))
%p A292506     end:
%p A292506 T:= n-> (p-> seq(coeff(p,x,i), i=0..n))(b(n$2)):
%p A292506 seq(T(n), n=0..12);
%t A292506 g[n_] := 2^(n-1) + If[OddQ[n], 0, Binomial[n, n/2]/2];
%t A292506 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 A292506 T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n}]][b[n, n]];
%t A292506 Table[T[n], {n, 0, 12}] // Flatten (* _Jean-François Alcover_, Jun 06 2018, from Maple *)
%Y A292506 Columns k=0-10 give: A000007, A027306 (for n>0), A316403, A316404, A316405, A316406, A316407, A316408, A316409, A316410, A316411.
%Y A292506 Row sums give A292548.
%Y A292506 T(2n,n) gives A292549.
%Y A292506 Cf. A209406, A226873, A290222.
%K A292506 nonn,tabl
%O A292506 0,5
%A A292506 _Alois P. Heinz_, Sep 17 2017