%I #14 Apr 30 2022 07:58:12

%S 1,1,4,8,25,53,148,328,858,1938,4862,11066,27042,61662,147774,336854,

%T 795678,1810466,4228330,9597694,22211897,50279985,115489274,260686018,

%U 594986149,1339215285,3040004744,6823594396,15416270130,34510814918,77644149076,173368564396

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

%H Alois P. Heinz, <a href="/A292548/b292548.txt">Table of n, a(n) for n = 0..3213</a>

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

%F Euler transform of A027306.

%e a(0) = 1: {}.

%e a(1) = 1: {1}.

%e a(2) = 4: {01}, {10}, {11}, {1,1}.

%e a(3) = 8: {011}, {101}, {110}, {111}, {1,01}, {1,10}, {1,11}, {1,1,1}.

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

%p a:= proc(n) option remember; `if`(n=0, 1, add(add(d*

%p g(d), d=numtheory[divisors](j))*a(n-j), j=1..n)/n)

%p end:

%p seq(a(n), n=0..35);

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

%t a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d*

%t g[d], {d, Divisors[j]}]*a[n-j], {j, 1, n}]/n];

%t Table[a[n], {n, 0, 35}] (* _Jean-François Alcover_, Apr 30 2022, after _Alois P. Heinz_ *)

%Y Row sums of A292506.

%Y Column k=2 of A292712.

%Y Cf. A027306.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Sep 18 2017