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A229812
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G.f. B(x) satisfies: B(x) = x + 2*A(x)*C(x), where A(x) = x + C(x)*B(x) and C(x) = x + 3*A(x)*B(x).
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3
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1, 2, 8, 34, 184, 1010, 5936, 35770, 221872, 1401890, 9005240, 58596754, 385519912, 2560003442, 17135913440, 115500024010, 783243325792, 5340020544962, 36581801008616, 251678879996290, 1738217149202584, 12046997299005938, 83759578272807440, 584052716966420698
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OFFSET
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1,2
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LINKS
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FORMULA
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G.f. B = B(x) satisfies:
(1) B = x + 2*x^2*(1+B)*(1+3*B)/(1-3*B^2)^2.
(2) B = x*(1+2*A)/(1-6*A^2) where A = x*(1+B)/(1-3*B^2) is the g.f. of A229811.
(3) B = x*(1+2*C)/(1-2*C^2) where C = x*(1+3*B)/(1-3*B^2) is the g.f. of A229813.
A*B*C = (A^2 - x*A) = (B^2 - x*B)/2 = (C^2 - x*C)/3.
a(n) ~ c*d^n/n^(3/2), where d = 7.438049365405038364... is the root of the equation -9 - 114*d - 442*d^2 - 792*d^3 - 660*d^4 - 432*d^5 - 192*d^6 - 24*d^7 + 8*d^8 = 0 and c = 0.08214781012909829230823825161142647948... - Vaclav Kotesovec, Sep 30 2013
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EXAMPLE
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G.f.: B(x) = x + 2*x^2 + 8*x^3 + 34*x^4 + 184*x^5 + 1010*x^6 + 5936*x^7 +...
Related series:
A(x) = x + x^2 + 5*x^3 + 23*x^4 + 121*x^5 + 673*x^6 + 3953*x^7 +...
C(x) = x + 3*x^2 + 9*x^3 + 45*x^4 + 225*x^5 + 1275*x^6 + 7389*x^7 +...
where B(x) = x + 2*A(x)*C(x).
(B(x)^2 - x*B(x))/2 = A(x)*B(x)*C(x) = x^3 + 6*x^4 + 33*x^5 + 192*x^6 + 1145*x^7 + 7038*x^8 + 44093*x^9 + 281232*x^10 + 1818513*x^11 + 11899830*x^12 +...
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PROG
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(PARI) {a(n)=local(A=x+x^2, B=x+2*x^2, C=x+3*x^2); for(i=1, n, A=x+B*C+x*O(x^n); B=x+2*A*C+x*O(x^n); C=x+3*A*B+x*O(x^n)); polcoeff(B, n)}
for(n=1, 30, print1(a(n), ", "))
(PARI) {a(n)=local(B=x); for(i=1, n, B=x+2*x^2*(1+B)*(1+3*B)/(1-3*B^2 +x*O(x^n))^2); polcoeff(B, n)}
for(n=1, 30, 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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