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%I #11 Dec 09 2015 17:49:19
%S 1,0,0,0,0,1,5,15,35,70,127,215,360,605,1080,2003,3890,7570,14715,
%T 27960,52255,95705,173295,311060,557400,999032,1795880,3235130,
%U 5835955,10521060,18931287,33956485,60692510,108087835,191883595,339724144,600203700,1058605775,1864535670,3279862975,5762287759,10109925380
%N G.f.: 1 / Product_{n>=0} (1 - x^(n+5))^((n+1)*(n+2)*(n+3)*(n+4)/4!).
%C Number of partitions of n objects of 5 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="/A264925/b264925.txt">Table of n, a(n) for n = 0..5000</a>
%F G.f.: exp( Sum_{n>=1} ( x^n/(1-x^n) )^5 /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)*(d-4)/4!.
%F a(n) ~ Pi^(95/288) / (2 * 3^(527/576) * 7^(239/1728) * n^(1103/1728)) * exp(-25*Zeta'(-1)/12 - log(2*Pi)/2 + 595*Zeta(3)/(48*Pi^2) - 29291*Zeta(5) / (128*Pi^4) - 2480625 * Zeta(3) * Zeta(5)^2 / (2*Pi^12) + 72930375 * Zeta(5)^3 / (2*Pi^14) - 1063324867500 * Zeta(5)^5/Pi^24 - 5*Zeta'(-3)/12 + (41 * 7^(1/6) * Pi/(768*sqrt(3)) - 2625 * sqrt(3) * 7^(1/6) * Zeta(3) * Zeta(5)/(2*Pi^7) + 540225 * sqrt(3) * 7^(1/6) * Zeta(5)^2/(16*Pi^9) - 4740474375 * sqrt(3) * 7^(1/6) * Zeta(5)^4/(4*Pi^19)) * n^(1/6) + (-25 * 7^(1/3) * Zeta(3)/(4*Pi^2) + 735 * 7^(1/3) * Zeta(5) /(8*Pi^4) - 3969000 * 7^(1/3) * Zeta(5)^3 / Pi^14) * n^(1/3) + (7*sqrt(7/3)*Pi/24 - 4725 * sqrt(21) * Zeta(5)^2 / Pi^9) * sqrt(n) - 45 * 7^(2/3) * Zeta(5)/(2*Pi^4) * n^(2/3) + 2*sqrt(3)*Pi / (5*7^(1/6)) * n^(5/6)). - _Vaclav Kotesovec_, Dec 09 2015
%e G.f.: A(x) = 1 + x^5 + 5*x^6 + 15*x^7 + 35*x^8 + 70*x^9 + 127*x^10 + 215*x^11 + 360*x^12 +...
%e where
%e 1/A(x) = (1-x^5) * (1-x^6)^5 * (1-x^7)^15 * (1-x^8)^35 * (1-x^9)^70 * (1-x^10)^126 * (1-x^11)^210 * (1-x^12)^330 * (1-x^13)^495 *...
%e Also,
%e log(A(x)) = (x/(1-x))^5 + (x^2/(1-x^2))^5/2 + (x^3/(1-x^3))^5/3 + (x^4/(1-x^4))^5/4 + (x^5/(1-x^5))^5/5 + (x^6/(1-x^6))^5/6 +...
%t nmax = 50; CoefficientList[Series[Product[1/(1-x^k)^((k-4)*(k-3)*(k-2)*(k-1)/24), {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)*(d-4)/4!)}
%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, A264924, A264926.
%Y Cf. A000391, A255052.
%K nonn
%O 0,7
%A _Paul D. Hanna_, Nov 28 2015
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