cp's OEIS Frontend

A316403 Number of multisets of exactly two nonempty binary words with a total of n letters such that no word has a majority of 0's.

%I A316403 #11 Jul 02 2018 16:29:02
%S A316403 1,3,10,23,59,134,320,699,1599,3434,7682,16246,35762,74892,163032,
%T A316403 338771,731051,1510466,3237206,6658530,14189790,29083988,61687496,
%U A316403 126076638,266332390,543061284,1143207236,2326521164,4882706596,9920514328,20764519984,42130081155
%N A316403 Number of multisets of exactly two nonempty binary words with a total of n letters such that no word has a majority of 0's.
%H A316403 Alois P. Heinz, <a href="/A316403/b316403.txt">Table of n, a(n) for n = 2..1000</a>
%F A316403 a(n) = [x^n y^2] 1/Product_{j>=1} (1-y*x^j)^A027306(j).
%e A316403 a(4) = 10: {1,011}, {1,101}, {1,110}, {1,111}, {01,01}, {01,10}, {01,11}, {10,10}, {10,11}, {11,11}.
%e A316403 a(5) = 23: {1,0011}, {1,0101}, {1,0110}, {1,0111}, {1,1001}, {1,1010}, {1,1011}, {1,1100}, {1,1101}, {1,1110}, {1,1111}, {01,011}, {01,101}, {01,110}, {01,111}, {10,011}, {10,101}, {10,110}, {10,111}, {11,011}, {11,101}, {11,110}, {11,111}.
%p A316403 g:= n-> 2^(n-1)+`if`(n::odd, 0, binomial(n, n/2)/2):
%p A316403 b:= proc(n, i) option remember; series(`if`(n=0 or i=1, x^n, add(
%p A316403        binomial(g(i)+j-1, j)*b(n-i*j, i-1)*x^j, j=0..n/i)), x, 3)
%p A316403     end:
%p A316403 a:= n-> coeff(b(n$2), x, 2):
%p A316403 seq(a(n), n=2..33);
%Y A316403 Column k=2 of A292506.
%Y A316403 Cf. A027306, A292549.
%K A316403 nonn
%O A316403 2,2
%A A316403 _Alois P. Heinz_, Jul 02 2018