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A274396
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G.f. A(x) satisfies: A(x)^3 - 3*A(x)^4 = A(x^3).
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2
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1, 1, 3, 12, 52, 246, 1224, 6300, 33300, 179643, 985014, 5473611, 30756638, 174460232, 997620618, 5744867987, 33286011182, 193909247415, 1135089596103, 6673249089811, 39385246833252, 233269982839202, 1386037324968892, 8259646628699352, 49352953593386343, 295622321452540572, 1774808912507223393, 10677836739621878302, 64367558119301035734, 388726705119505499253, 2351595589979028349894
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OFFSET
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1,3
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LINKS
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FORMULA
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G.f. A(x) satisfies: A(B(x)^3) = x^3 - 3*x^4, where A(B(x)) = x.
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EXAMPLE
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G.f.: A(x) = x + x^2 + 3*x^3 + 12*x^4 + 52*x^5 + 246*x^6 + 1224*x^7 + 6300*x^8 + 33300*x^9 + 179643*x^10 + 985014*x^11 + 5473611*x^12 + 30756638*x^13 + 174460232*x^14 + 997620618*x^15 + 5744867987*x^16 + 33286011182*x^17 +...
such that A(x)^3 - 3*A(x)^4 = A(x^3).
RELATED SERIES.
A(x)^3 = x^3 + 3*x^4 + 12*x^5 + 55*x^6 + 264*x^7 + 1329*x^8 + 6915*x^9 + 36846*x^10 + 200100*x^11 + 1103367*x^12 + 6160500*x^13 + 34758036*x^14 + 197858104*x^15 + 1134939672*x^16 + 6553596816*x^17 + 38064666342*x^18 +...
A(x)^4 = x^4 + 4*x^5 + 18*x^6 + 88*x^7 + 443*x^8 + 2304*x^9 + 12282*x^10 + 66700*x^11 + 367785*x^12 + 2053500*x^13 + 11586012*x^14 + 65952684*x^15 + 378313224*x^16 + 2184532272*x^17 + 12688222032*x^18 +...
where
A(x)^3 - 3*A(x)^4 = x^3 + x^6 + 3*x^9 + 12*x^12 + 52*x^15 + 246*x^18 + 1224*x^21 +...
Let B(x) be the series reversion of g.f. A(x), so that A(B(x)) = x, then
B(x) = x - x^2 - x^3 - 2*x^4 - 2*x^5 - 8*x^6 - 18*x^7 - 39*x^8 - 111*x^9 - 266*x^10 - 683*x^11 - 1859*x^12 - 4824*x^13 - 13091*x^14 - 35642*x^15 - 96917*x^16 - 267206*x^17 - 738605*x^18 - 2048769*x^19 - 5720874*x^20 - 16009584*x^21 -...
where B(x^3 - 3*x^4) = B(x)^3.
Further, we have
A(B(x)^3) = C(x) = x^3 - 3*x^4,
A(B(x)^9) = C(C(x)) = x^9 - 9*x^10 + 27*x^11 - 30*x^12 + 36*x^13 - 162*x^14 + 324*x^15 - 243*x^16, ...
so that A(B(x)^(3^n)) = C^n(x), the n-th iteration of C(x) = x^3 - 3*x^4.
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PROG
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(PARI) /* From A(B(x)^3) = x^3 - 3*x^4, where A(B(x)) = x: */
{a(n) = my(A=[1, 1], F, B); for(i=1, n, A=concat(A, 0); F=x*Ser(A); B=serreverse(F); A[#A] = Vec(subst(F, x, B^3))[#A]/3); A[n]}
for(n=1, 40, print1(a(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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