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A183038
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G.f.: exp( Sum_{n>=1} A051064(n)*3^A051064(n)*x^n/n ) where A051064(n) equals the 3-adic valuation of 3n.
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3
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1, 3, 6, 15, 30, 51, 93, 156, 240, 387, 597, 870, 1311, 1920, 2697, 3873, 5448, 7422, 10278, 14016, 18636, 25098, 33402, 43548, 57333, 74757, 95820, 123780, 158637, 200391, 254778, 321798, 401451, 503490, 627915, 774726, 960156, 1184205, 1446873
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OFFSET
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0,2
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COMMENTS
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Compare to B(x), the g.f. of the number of partitions of 3n into powers of 3 (A005704):
B(x) = exp( Sum_{n>=1} 3^A051064(n)*x^n/n ) = (1-x)^(-1)*Product_{n>=0} 1/(1 - x^(3^n)).
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LINKS
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FORMULA
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G.f. satisfies: A(x) = (1-x^3)/(1-x)^3 * A(x^3)^2/A(x^9).
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.
Define TRISECTIONS: A(x) = T_0(x^3) + x*T_1(x^3) + x^2*T_2(x^3), then:
T_1(x)/T_0(x) = 3*(1 + 2*x)/(1 + 7*x + x^2) and
T_2(x)/T_0(x) = 3*(2 + x)/(1 + 7*x + x^2).
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EXAMPLE
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G.f.: A(x) = 1 + 3*x + 6*x^2 + 15*x^3 + 30*x^4 + 51*x^5 + 93*x^6 +...
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 +...
G.f. satisfies A(x) = A(x^3)*G(x) where G(x) is the g.f. of A161809:
G(x) = 1 + 3*x + 6*x^2 + 12*x^3 + 21*x^4 + 33*x^5 + 51*x^6 +...
TRISECTIONS of g.f. begin:
T_0(x) = 1 + 15*x + 93*x^2 + 387*x^3 + 1311*x^4 + 3873*x^5 +...
T_1(x) = 3 + 30*x + 156*x^2 + 597*x^3 + 1920*x^4 + 5448*x^5 +...
T_2(x) = 6 + 51*x + 240*x^2 + 870*x^3 + 2697*x^4 + 7422*x^5 +...
where the ratios involve Fibonacci numbers:
T_1(x)/T_0(x) = 3*(1 - 5*x + 34*x^2 - 233*x^3 +...+ (-1)^n*Fibonacci(4n+1)*x^n +...);
T_2(x)/T_0(x) = 3*(2 - 13*x + 89*x^2 - 610*x^3 +...+ (-1)^n*Fibonacci(4n+3)*x^n +...).
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PROG
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(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)}
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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