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%I #14 Dec 09 2015 18:02:03
%S 1,0,0,0,0,0,1,6,21,56,126,252,463,798,1329,2184,3696,6552,12405,
%T 24486,49524,99722,197967,383796,727609,1350174,2466534,4457844,
%U 8022819,14448168,26142810,47603010,87222576,160522228,295996791,545445468,1002392105,1834644210,3342375099,6061611192,10949981496,19720143366,35440268956
%N G.f.: 1 / Product_{n>=0} (1 - x^(n+6))^((n+1)*(n+2)*(n+3)*(n+4)*(n+5)/5!).
%C Number of partitions of n objects of 6 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="/A264926/b264926.txt">Table of n, a(n) for n = 0..5000</a>
%F G.f.: exp( Sum_{n>=1} ( x^n/(1-x^n) )^6 /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)*(d-5)/5!.
%F a(n) ~ (3*Zeta(7))^(11153/423360) / (2^(200527/423360) * n^(222833/423360) * sqrt(7*Pi)) * exp(137*Zeta'(-1)/60 + log(2*Pi)/2 + 15*Zeta(3) / (32*Pi^2) - 3*Zeta(5) / (32*Pi^4) - Pi^36 / (102098378167208640 * Zeta(7)^5) + 17 * Pi^24 * Zeta(5) / (571643448768 * Zeta(7)^4) - Pi^22 / (9073705536 * Zeta(7)^3) - 289 * Pi^12 * Zeta(5)^2 / (12002256 * Zeta(7)^3) + 137 * Pi^12 * Zeta(3) / (60011280 * Zeta(7)^2) + 17 * Pi^10 * Zeta(5) / (127008 * Zeta(7)^2) + 4913 * Zeta(5)^3 / (1512 * Zeta(7)^2) - 253 * Pi^8 / (1016064 * Zeta(7)) - 2329 * Zeta(3) * Zeta(5) / (1260 * Zeta(7)) + Zeta'(-5)/120 + 17 * Zeta'(-3)/24 + (-11*Pi^30 / (1544080410553464 * 6^(1/7) * Zeta(7)^(29/7)) + 85 * Pi^18 * Zeta(5) / (4631370534 * 6^(1/7) * Zeta(7)^(22/7)) - Pi^16 / (14002632 * 6^(1/7) * Zeta(7)^(15/7)) - 289 * Pi^6 * Zeta(5)^2 / (27783 * 6^(1/7) * Zeta(7)^(15/7)) + 137 * Pi^6 * Zeta(3) / (79380 * 6^(1/7) * Zeta(7)^(8/7)) + 17 * Pi^4 * Zeta(5) / (336 * 6^(1/7) * Zeta(7)^(8/7)) - Pi^2 / (6^(8/7) * Zeta(7)^(1/7))) * n^(1/7) + (-Pi^24 / (194517562428 * 6^(2/7) * Zeta(7)^(23/7)) + 17 * Pi^12 * Zeta(5) / (1555848 * 6^(2/7) * Zeta(7)^(16/7)) - Pi^10 / (21168 * 6^(2/7) * Zeta(7)^(9/7)) - 289 * Zeta(5)^2 / (84 * 6^(2/7) * Zeta(7)^(9/7)) + 137 * Zeta(3) / (60 * (6*Zeta(7))^(2/7))) * n^(2/7) + (-5*Pi^18 / (1323248724 * 6^(3/7) * Zeta(7)^(17/7)) + 17 * Pi^6 * Zeta(5) / (2646 * 6^(3/7) * Zeta(7)^(10/7)) - Pi^4 /(24 * (6*Zeta(7))^(3/7))) * n^(3/7) + (-Pi^12 / (333396 * 6^(4/7) * Zeta(7)^(11/7)) + 17 * Zeta(5) / (4 * (6*Zeta(7))^(4/7))) * n^(4/7) - Pi^6 / (315*(6*Zeta(7))^(5/7)) * n^(5/7) + 7 * Zeta(7)^(1/7) / 6^(6/7) * n^(6/7)). - _Vaclav Kotesovec_, Dec 09 2015
%e G.f.: A(x) = 1 + x^6 + 6*x^7 + 21*x^8 + 56*x^9 + 126*x^10 + 252*x^11 + 463*x^12 +...
%e where
%e 1/A(x) = (1-x^6) * (1-x^7)^6 * (1-x^8)^21 * (1-x^9)^56 * (1-x^10)^126 * (1-x^11)^252 * (1-x^12)^462 * (1-x^13)^792 * (1-x^14)^1287 * (1-x^15)^2002 *...
%e Also,
%e log(A(x)) = (x/(1-x))^6 + (x^2/(1-x^2))^6/2 + (x^3/(1-x^3))^6/3 + (x^4/(1-x^4))^6/4 + (x^5/(1-x^5))^6/5 + (x^6/(1-x^6))^6/6 +...
%t nmax = 50; CoefficientList[Series[Product[1/(1-x^k)^((k-5)*(k-4)*(k-3)*(k-2)*(k-1)/120), {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+6) +x*O(x^n) )^((k+1)*(k+2)*(k+3)*(k+4)*(k+5)/5!) ); 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))^6 /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)*(d-5)/5! )}
%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, A264925.
%Y Cf. A000417.
%K nonn
%O 0,8
%A _Paul D. Hanna_, Nov 28 2015
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