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A274271 Number of partitions of 3^n into at most four parts.

%I A274271 #12 Jun 21 2016 09:29:05
%S A274271 1,3,18,225,4410,105903,2746098,73140525,1965803130,52995903003,
%T A274271 1430162760978,38607856205625,1042353276205050,28143008896575303,
%U A274271 759856474192364658,20516081909157771525,553933825501236490170,14956209814120079146803,403817633711525094117138
%N A274271 Number of partitions of 3^n into at most four parts.
%H A274271 Colin Barker, <a href="/A274271/b274271.txt">Table of n, a(n) for n = 0..650</a>
%F A274271 Coefficient of x^(3^n) in 1/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)).
%F A274271 Conjectures (Start)
%F A274271 a(n) = ((3+3^n)^2*(9+3^n))/144 for n>1.
%F A274271 a(n) = 40*a(n-1)-390*a(n-2)+1080*a(n-3)-729*a(n-4) for n>4.
%F A274271 G.f.: (1-37*x+288*x^2-405*x^3-81*x^4) / ((1-x)*(1-3*x)*(1-9*x)*(1-27*x)).
%F A274271 (End)
%o A274271 (PARI)
%o A274271 \\ b(n) is the coefficient of x^n in the g.f. 1/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)).
%o A274271 b(n) = round(real(68+36*(-1)^n+18*((-I)^n+I^n)+(16*exp(-2/3*I*n*Pi)*(1+I*sqrt(3)+2*exp((4*I*n*Pi)/3)))/(1+(-1)^(1/3))+59*(1+n)+9*(-1)^n*(1+n)+18*(1+n)*(2+n)+2*(1+n)*(2+n)*(3+n))/288)
%o A274271 vector(20, n, n--; b(3^n))
%Y A274271 A subsequence of A001400.
%Y A274271 Cf. A274100 (2^n), A274272 (5^n).
%K A274271 nonn
%O A274271 0,2
%A A274271 _Colin Barker_, Jun 17 2016