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A265257 Number of odd singletons in all partitions of n (n>=0).

%I A265257 #11 Jan 01 2016 22:13:40
%S A265257 0,1,0,2,2,5,5,11,13,23,28,45,57,86,108,156,199,276,350,475,601,798,
%T A265257 1005,1312,1646,2120,2643,3365,4178,5264,6500,8122,9981,12375,15136,
%U A265257 18638,22697,27779,33679,40993,49504,59947,72109
%N A265257 Number of odd singletons in all partitions of n (n>=0).
%H A265257 Alois P. Heinz, <a href="/A265257/b265257.txt">Table of n, a(n) for n = 0..1000</a>
%F A265257 a(n) = Sum(k*A265255(n,k), k>=0).
%F A265257 G.f.: g(x) = x(1 - x + x^2)/((1-x^4)*Product_{j>=1}(1-x^j)).
%F A265257 From _Vaclav Kotesovec_, Jan 01 2016: (Start)
%F A265257 a(n) = 1/4 * A000070(n) - 3/4 * A087787(n) + 1/2 * A092295(n).
%F A265257 a(n) ~ exp(Pi*sqrt(2*n/3)) / (8*Pi*sqrt(2*n)).
%F A265257 (End)
%e A265257 a(6) = 5 because in [1,1,1,3], [1,2,3], [1,5] we have 1+2+2 odd singletons, while the other 8 partitions of 6 have no odd singletons.
%p A265257 g := x*(1-x+x^2)/((1-x^4)*mul(1-x^j, j = 1 .. 80)): gser := series(g, x = 0, 55): seq(coeff(gser, x, m), m = 0 .. 50);
%p A265257 # second Maple program:
%p A265257 b:= proc(n, i) option remember; `if`(n=0, [1, 0],
%p A265257       `if`(i<1, 0, add((p-> `if`(j=1 and i::odd, p+
%p A265257       [0, p[1]], p))(b(n-i*j, i-1)), j=0..n/i)))
%p A265257     end:
%p A265257 a:= n-> b(n$2)[2]:
%p A265257 seq(a(n), n=0..80);  # _Alois P. Heinz_, Jan 01 2016
%t A265257 nmax = 50; CoefficientList[Series[x*(1-x+x^2)/(1-x^4) * Product[1/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Jan 01 2016 *)
%Y A265257 Cf. A265255.
%K A265257 nonn
%O A265257 0,4
%A A265257 _Emeric Deutsch_, Jan 01 2016