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Number of odd singletons in all partitions of n (n>=0).
3

%I #11 Jan 01 2016 22:13:40

%S 0,1,0,2,2,5,5,11,13,23,28,45,57,86,108,156,199,276,350,475,601,798,

%T 1005,1312,1646,2120,2643,3365,4178,5264,6500,8122,9981,12375,15136,

%U 18638,22697,27779,33679,40993,49504,59947,72109

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

%H Alois P. Heinz, <a href="/A265257/b265257.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) = Sum(k*A265255(n,k), k>=0).

%F G.f.: g(x) = x(1 - x + x^2)/((1-x^4)*Product_{j>=1}(1-x^j)).

%F From _Vaclav Kotesovec_, Jan 01 2016: (Start)

%F a(n) = 1/4 * A000070(n) - 3/4 * A087787(n) + 1/2 * A092295(n).

%F a(n) ~ exp(Pi*sqrt(2*n/3)) / (8*Pi*sqrt(2*n)).

%F (End)

%e 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 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 # second Maple program:

%p b:= proc(n, i) option remember; `if`(n=0, [1, 0],

%p `if`(i<1, 0, add((p-> `if`(j=1 and i::odd, p+

%p [0, p[1]], p))(b(n-i*j, i-1)), j=0..n/i)))

%p end:

%p a:= n-> b(n$2)[2]:

%p seq(a(n), n=0..80); # _Alois P. Heinz_, Jan 01 2016

%t 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 Cf. A265255.

%K nonn

%O 0,4

%A _Emeric Deutsch_, Jan 01 2016