|
%I #17 Dec 09 2015 16:51:23
%S 1,0,0,1,3,6,11,18,33,57,105,183,330,567,990,1693,2904,4917,8343,
%T 14010,23511,39171,65100,107592,177352,290931,475905,775381,1259637,
%U 2039094,3291613,5296467,8499339,13599292,21702795,34541724,54839894,86847255,137212197,216274466,340129773,533726442,835732774,1305877914,2036369010
%N G.f.: 1 / Product_{n>=0} (1 - x^(n+3))^((n+1)*(n+2)/2!).
%C Number of partitions of n objects of 3 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="/A264923/b264923.txt">Table of n, a(n) for n = 0..6000</a>
%F G.f.: exp( Sum_{n>=1} ( x^n/(1-x^n) )^3 /n ).
%F G.f.: exp( Sum_{n>=1} L(n) * x^n/n ), where L(n) = Sum_{d|n} d*(d-1)*(d-2)/2!.
%F a(n) ~ Pi^(3/8) / (2^(55/32) * 15^(7/32) * n^(23/32)) * exp(29*Zeta(3)/(8*Pi^2) - log(2*Pi)/2 - 3*Zeta'(-1)/2 - 2025*Zeta(3)^3/(2*Pi^8) + (5^(1/4)*Pi/6^(3/4) - 135*15^(1/4)*Zeta(3)^2/(2^(7/4)*Pi^5)) * n^(1/4) - 3*sqrt(15*n/2)*Zeta(3)/Pi^2 + 2^(7/4)*Pi/(3*15^(1/4)) * n^(3/4)). - _Vaclav Kotesovec_, Dec 09 2015
%e G.f.: A(x) = 1 + x^3 + 3*x^4 + 6*x^5 + 11*x^6 + 18*x^7 + 33*x^8 + 57*x^9 + 105*x^10 +...
%e where
%e 1/A(x) = (1-x^3) * (1-x^4)^3 * (1-x^5)^6 * (1-x^6)^10 * (1-x^7)^15 * (1-x^8)^21 * (1-x^9)^28 * (1-x^10)^36 * (1-x^11)^45 *...
%e Also,
%e log(A(x)) = (x/(1-x))^3 + (x^2/(1-x^2))^3/2 + (x^3/(1-x^3))^3/3 + (x^4/(1-x^4))^3/4 + (x^5/(1-x^5))^3/5 + (x^6/(1-x^6))^3/6 +...
%t nmax = 50; CoefficientList[Series[Product[1/(1-x^k)^((k-2)*(k-1)/2), {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+3) +x*O(x^n) )^((k+1)*(k+2)/2) ); 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))^3 /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)/2! )}
%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, A264924, A264925, A264926.
%Y Cf. A000294, A217093, A258349.
%K nonn
%O 0,5
%A _Paul D. Hanna_, Nov 28 2015
|
