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A292549 Number of multisets of exactly n nonempty binary words with a total of 2n letters such that no word has a majority of 0's.

%I A292549 #14 May 11 2019 10:11:05
%S A292549 1,3,10,33,98,291,826,2320,6342,17188,45750,120733,314690,813854,
%T A292549 2085363,5306878,13406382,33665476,84031608,208655086,515469203,
%U A292549 1267600993,3103490884,7567559622,18381579206,44487740012,107301636460,257967350824,618279370985
%N A292549 Number of multisets of exactly n nonempty binary words with a total of 2n letters such that no word has a majority of 0's.
%H A292549 Alois P. Heinz, <a href="/A292549/b292549.txt">Table of n, a(n) for n = 0..3163</a>
%F A292549 a(n) = A292506(2n,n) = A292506(2n+j,n+j) for j >= 0.
%F A292549 G.f.: Product_{j>=1} 1/(1-x^j)^A027306(j+1).
%F A292549 Euler transform of j-> A027306(j+1).
%e A292549 a(0) = 1: {}.
%e A292549 a(1) = 3: {01}, {10}, {11}.
%e A292549 a(2) = 10: {1,011}, {1,101}, {1,110}, {1,111}, {01,01}, {01,10}, {01,11}, {10,10}, {10,11}, {11,11}.
%p A292549 g:= n-> 2^(n-1)+`if`(n::odd, 0, binomial(n, n/2)/2):
%p A292549 a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
%p A292549       g(d+1), d=numtheory[divisors](j))*a(n-j), j=1..n)/n)
%p A292549     end:
%p A292549 seq(a(n), n=0..35);
%t A292549 g[n_] := 2^(n-1) + If[OddQ[n], 0, Binomial[n, n/2]/2];
%t A292549 a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d*g[d+1], {d, Divisors[j]}]*a[n-j], {j, 1, n}]/n];
%t A292549 Table[a[n], {n, 0, 35}] (* _Jean-François Alcover_, May 11 2019, after _Alois P. Heinz_ *)
%Y A292549 Cf. A292506.
%K A292549 nonn
%O A292549 0,2
%A A292549 _Alois P. Heinz_, Sep 18 2017