A183038 - OEIS

%I #10 Mar 30 2012 18:37:23

%S 1,3,6,15,30,51,93,156,240,387,597,870,1311,1920,2697,3873,5448,7422,

%T 10278,14016,18636,25098,33402,43548,57333,74757,95820,123780,158637,

%U 200391,254778,321798,401451,503490,627915,774726,960156,1184205,1446873

%N G.f.: exp( Sum_{n>=1} A051064(n)*3^A051064(n)*x^n/n ) where A051064(n) equals the 3-adic valuation of 3n.

%C Compare to B(x), the g.f. of the number of partitions of 3n into powers of 3 (A005704):

%C B(x) = exp( Sum_{n>=1} 3^A051064(n)*x^n/n ) = (1-x)^(-1)*Product_{n>=0} 1/(1 - x^(3^n)).

%F G.f. satisfies: A(x) = (1-x^3)/(1-x)^3 * A(x^3)^2/A(x^9).

%F G.f. satisfies: A(x) = A(x^3)*G(x) where G(x) = G(x^3)*(1+x+x^2)/(1-x)^2 is the g.f. of A161809.

%F Define TRISECTIONS: A(x) = T_0(x^3) + x*T_1(x^3) + x^2*T_2(x^3), then:

%F T_1(x)/T_0(x) = 3*(1 + 2*x)/(1 + 7*x + x^2) and

%F T_2(x)/T_0(x) = 3*(2 + x)/(1 + 7*x + x^2).

%e G.f.: A(x) =  1 + 3*x + 6*x^2 + 15*x^3 + 30*x^4 + 51*x^5 + 93*x^6 +...

%e log(A(x)) = 3*x + 3*x^2/2 + 18*x^3/3 + 3*x^4/4 + 3*x^5/5 + 18*x^6/6 + 3*x^7/7 + 3*x^8/8 + 81*x^9/9 + 3*x^10/10 + 3*x^11/11 + 18*x^12/12 +...

%e G.f. satisfies A(x) = A(x^3)*G(x) where G(x) is the g.f. of A161809:

%e G(x) = 1 + 3*x + 6*x^2 + 12*x^3 + 21*x^4 + 33*x^5 + 51*x^6 +...

%e TRISECTIONS of g.f. begin:

%e T_0(x) = 1 + 15*x + 93*x^2 + 387*x^3 + 1311*x^4 + 3873*x^5 +...

%e T_1(x) = 3 + 30*x + 156*x^2 + 597*x^3 + 1920*x^4 + 5448*x^5 +...

%e T_2(x) = 6 + 51*x + 240*x^2 + 870*x^3 + 2697*x^4 + 7422*x^5 +...

%e where the ratios involve Fibonacci numbers:

%e T_1(x)/T_0(x) = 3*(1 - 5*x + 34*x^2 - 233*x^3 +...+ (-1)^n*Fibonacci(4n+1)*x^n +...);

%e T_2(x)/T_0(x) = 3*(2 - 13*x + 89*x^2 - 610*x^3 +...+ (-1)^n*Fibonacci(4n+3)*x^n +...).

%o (PARI) {a(n)=polcoeff(exp(sum(m=1,n,valuation(3*m,3)*3^valuation(3*m,3)*x^m/m)+x*O(x^n)),n)}

%Y Cf. A161809, A183039, A051064, A005704, A183036.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Dec 19 2010