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A130126 Number of partitions of n in which each even part has odd multiplicity.

%I A130126 #14 Jan 03 2016 09:37:49
%S A130126 1,1,2,3,4,6,10,13,17,24,33,43,58,75,98,127,161,205,262,328,414,517,
%T A130126 641,794,982,1205,1475,1803,2197,2664,3230,3896,4693,5640,6754,8077,
%U A130126 9647,11479,13637,16178,19152,22624,26695,31426,36948,43372,50819,59463,69490
%N A130126 Number of partitions of n in which each even part has odd multiplicity.
%H A130126 Alois P. Heinz, <a href="/A130126/b130126.txt">Table of n, a(n) for n = 0..1000</a>
%F A130126 G.f.: Product_{n>=1} (1+q^(2n)-q^(4n))/((1-q^(2n-1))(1-q^(4n))).
%F A130126 a(n) ~ sqrt(Pi^2/2 + 4*log(phi)^2) * exp(sqrt((Pi^2 + 8*log(phi)^2)*(n/2))) / (4*Pi*n), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - _Vaclav Kotesovec_, Jan 03 2016
%e A130126 a(5) = 6 because we have 5, 41, 32, 311, 2111 and 11111 (221 does not qualify).
%p A130126 g:=product((1+q^(2*n)-q^(4*n))/((1-q^(2*n-1))*(1-q^(4*n))),n=1..50): gser:= series(g,q=0,45): seq(coeff(gser,q,n),n=0..42); # _Emeric Deutsch_, Aug 24 2007
%p A130126 # second Maple program:
%p A130126 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, b(n, i-1)+
%p A130126       add(`if`(irem(i, 2)=0 and irem(j, 2)=0, 0, b(n-i*j, i-1)), j=1..n/i)))
%p A130126     end:
%p A130126 a:= n-> b(n, n):
%p A130126 seq(a(n), n=0..50);  # _Alois P. Heinz_, Feb 27 2013
%t A130126 b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1] + Sum[If[Mod[i, 2] == 0 && Mod[j, 2] == 0, 0, b[n - i*j, i - 1]], {j, 1, n/i}]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 50}] (* _Jean-François Alcover_, Jan 24 2014, after _Alois P. Heinz_ *)
%t A130126 nmax = 50; CoefficientList[Series[Product[(1 + x^(2*k) - x^(4*k))/((1-x^(2*k-1)) * (1-x^(4*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Jan 03 2016 *)
%Y A130126 Cf. A131942.
%K A130126 easy,nonn
%O A130126 0,3
%A A130126 _Vladeta Jovovic_, Aug 01 2007
%E A130126 More terms from _Emeric Deutsch_, Aug 24 2007