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Search: id:A140093
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| A140093 |
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G.f. satisfies: A(x - x^3) = x^3 - x^9. |
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1
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| 1, 0, 3, 0, 12, 0, 54, 0, 264, 0, 1365, 0, 7344, 0, 40698, 0, 230736, 0, 1332045, 0, 7803900, 0, 46280520, 0, 277294752, 0, 1676056044, 0, 10207492512, 0, 62576750610, 0, 385856498592, 0, 2391506423565, 0, 14890534347780, 0
(list; graph; listen)
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OFFSET
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3,3
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FORMULA
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a(2n)=0; a(2n+1) = 6*C(3*n-3,n-2)/n for n>1 with a(1)=0, a(3)=1.
G.f.: A(x) = x^3 + 3*x^5*G(x^2)^4 where G(x) = 1 + x*G(x)^3 = g.f. of A001764.
Define A_{n+1}(x) = A( A_{n}(x) ) with A_0(x)=x, then
A_{n}(x) = (x*G(x^2))^(3^n) - (x*G(x^2))^(3^(n+1)) and so
Sum_{n>=0} A_{n}(x) = x*G(x^2) where G(x) = g.f. of A001764.
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EXAMPLE
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A(x) = x^3 + 3*x^5 + 12*x^7 + 54*x^9 + 264*x^11 + 1365*x^13 +...
x^3 - x^9 = (x-x^3)^3 + 3*(x-x^3)^5 + 12*(x-x^3)^7 + 54*(x-x^3)^9 +...
Let G(x) = 1 + x*G(x)^3 = g.f. of A001764, then
A(x) = (x*G(x^2))^3 - (x*G(x^2))^9 ;
A(A(x)) = (x*G(x^2))^9 - (x*G(x^2))^27 ;
A(A(A(x))) = (x*G(x^2))^27 - (x*G(x^2))^81 ; etc.
Related expansions are:
G(x) = 1 + x + 3*x^2 + 12*x^3 + 55*x^4 + 273*x^5 + 1428*x^6 +...
A(A(x)) = x^9 + 9*x^11 + 63*x^13 + 408*x^15 + 2565*x^17 +...
A(A(A(x))) = x^27 + 27*x^29 + 432*x^31 + 5355*x^33 + 56943*x^35 +...
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PROGRAM
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(PARI) a(n)=if(n<3|n%2==0, 0, if(n==3, 1, 12*binomial(3*(n-1)/2-3, (n-1)/2-2)/(n-1)))
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CROSSREFS
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Cf A001764 (enumerates ternary trees).
Sequence in context: A019264 A028851 A132221 this_sequence A055314 A110890 A071534
Adjacent sequences: A140090 A140091 A140092 this_sequence A140094 A140095 A140096
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KEYWORD
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nonn
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AUTHOR
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Paul D. Hanna (pauldhanna(AT)juno.com), May 11 2008
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