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A116449 Number of partitions of n into an equal number of prime and composite parts.

%I A116449 #14 Dec 30 2017 09:50:41
%S A116449 1,0,0,0,0,0,1,1,1,2,1,4,3,4,4,6,6,11,11,13,17,20,26,32,37,43,55,63,
%T A116449 81,92,111,126,159,183,225,259,307,357,430,497,589,683,797,929,1093,
%U A116449 1270,1478,1712,1979,2303,2665,3086,3556,4102,4716,5444,6256,7194,8243,9456,10824
%N A116449 Number of partitions of n into an equal number of prime and composite parts.
%C A116449 a(n) = A002865(n) - A116450(n).
%H A116449 Andrew Howroyd, <a href="/A116449/b116449.txt">Table of n, a(n) for n = 0..1000</a>
%e A116449 a(14) = #{2+2*2*3, (2+2)+(2*3+2*2), 5+3*3, (3+3)+(2*2+2*2)} =
%e A116449 4;
%e A116449 a(15) = #{3+2*2*3, 5+2*5, (2+3)+(2*2+2*3), 7+2*2*2,
%e A116449 (2+5)+(2*2+2*2), 11+2*2} = 6.
%t A116449 terms = 61;
%t A116449 pp = 1/(Product[1 - If[PrimeQ[k], y, 1/y]*x^k, {k, 2, terms-3}] + O[x]^terms) // Normal;
%t A116449 Take[Expand[pp ], terms-5] // CoefficientList[#, x]& (* _Jean-François Alcover_, Dec 30 2017, after _Andrew Howroyd_ *)
%o A116449 (PARI)
%o A116449 parts(n)={1/(prod(k=2, n, 1 - if(isprime(k), y, 1/y)*x^k + O(x*x^n)))}
%o A116449 {my(n=60); apply(p->polcoeff(p,0), Vec(parts(n)))} \\ _Andrew Howroyd_, Dec 29 2017
%K A116449 nonn
%O A116449 0,10
%A A116449 _Reinhard Zumkeller_, Feb 16 2006
%E A116449 a(0)=1 from _Andrew Howroyd_, Dec 29 2017