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%I #12 Dec 09 2015 16:51:43
%S 1,0,0,0,1,4,10,20,36,60,104,180,336,620,1174,2160,3961,7100,12690,
%T 22424,39651,69820,122970,215904,378470,660872,1150740,1996200,
%U 3452685,5952916,10237576,17559460,30049285,51301020,87390872,148534232,251916041,426329040,720003646,1213481344,2041155052,3426721080
%N G.f.: 1 / Product_{n>=0} (1 - x^(n+4))^((n+1)*(n+2)*(n+3)/3!).
%C Number of partitions of n objects of 4 colors, where each part must contain at least one of each color. [Conjecture - see comment by Franklin T. Adams-Watters in A052847].
%H Vaclav Kotesovec, <a href="/A264924/b264924.txt">Table of n, a(n) for n = 0..5000</a>
%F G.f.: exp( Sum_{n>=1} ( x^n/(1-x^n) )^4 /n ).
%F G.f.: exp( Sum_{n>=1} L(n) * x^n/n ), where L(n) = Sum_{d|n} d*(d-1)*(d-2)*(d-3)/3!.
%F a(n) ~ Zeta(5)^(109/3600) / (2^(791/1800) * n^(1909/3600) * sqrt(5*Pi)) * exp(11*Zeta'(-1)/6 + log(2*Pi)/2 + Zeta(3)/(4*Pi^2) - Pi^16/(194400000 * Zeta(5)^3) + 11*Pi^8 * Zeta(3)/(108000 * Zeta(5)^2) - Pi^6/(1800*Zeta(5)) - 121*Zeta(3)^2/(360*Zeta(5)) + Zeta'(-3)/6 + (-Pi^12/(1350000 * 2^(2/5) * Zeta(5)^(11/5)) + 11*Pi^4 * Zeta(3)/(900 * 2^(2/5) * Zeta(5)^(6/5)) - Pi^2/(3*2^(7/5) * Zeta(5)^(1/5))) * n^(1/5) + (-Pi^8/(9000 * 2^(4/5) * Zeta(5)^(7/5)) + 11*Zeta(3)/(3*2^(9/5) * Zeta(5)^(2/5))) * n^(2/5) - Pi^4/(90 * 2^(1/5) * Zeta(5)^(3/5)) * n^(3/5) + 5*Zeta(5)^(1/5) / 2^(8/5) * n^(4/5)). - _Vaclav Kotesovec_, Dec 09 2015
%e G.f.: A(x) = 1 + x^4 + 4*x^5 + 10*x^6 + 20*x^7 + 36*x^8 + 60*x^9 + 104*x^10 + 180*x^11 +...
%e where
%e 1/A(x) = (1-x^4) * (1-x^5)^4 * (1-x^6)^10 * (1-x^7)^20 * (1-x^8)^35 * (1-x^9)^56 * (1-x^10)^84 * (1-x^11)^120 * (1-x^12)^165 *...
%e Also,
%e log(A(x)) = (x/(1-x))^4 + (x^2/(1-x^2))^4/2 + (x^3/(1-x^3))^4/3 + (x^4/(1-x^4))^4/4 + (x^5/(1-x^5))^4/5 + (x^6/(1-x^6))^4/6 +...
%t nmax = 50; CoefficientList[Series[Product[1/(1-x^k)^((k-3)*(k-2)*(k-1)/6), {k,1,nmax}], {x,0,nmax}], x] (* _Vaclav Kotesovec_, Dec 09 2015 *)
%o (PARI) {a(n) = my(A=1); A = prod(k=0,n, 1/(1 - x^(k+4) +x*O(x^n) )^((k+1)*(k+2)*(k+3)/3!) ); polcoeff(A,n)}
%o for(n=0,50,print1(a(n),", "))
%o (PARI) {a(n) = my(A=1); A = exp( sum(k=1,n+1, (x^k/(1 - x^k))^4 /k +x*O(x^n) ) ); polcoeff(A,n)}
%o for(n=0,50,print1(a(n),", "))
%o (PARI) {L(n) = sumdiv(n,d, d*(d-1)*(d-2)*(d-3)/3! )}
%o {a(n) = my(A=1); A = exp( sum(k=1,n+1, L(k) * x^k/k +x*O(x^n) ) ); polcoeff(A,n)}
%o for(n=0,50,print1(a(n),", "))
%Y Cf. A052847, A264923, A264925, A264926.
%Y Cf. A000335, A255050, A258352.
%K nonn
%O 0,6
%A _Paul D. Hanna_, Nov 28 2015
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