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A323692
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G.f. satisfies: A(x) = x + A( A(x)^3 + A(x)^4 ).
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1
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1, 0, 1, 1, 3, 7, 16, 45, 111, 311, 834, 2329, 6521, 18429, 52667, 151095, 437178, 1270035, 3710065, 10882077, 32044740, 94700739, 280749180, 834793837, 2488822697, 7438604115, 22283235185, 66893731444, 201208674387, 606321286160, 1830213820180, 5533440540954, 16754840359013, 50803933761199, 154251935227044, 468929198610654, 1427240650197467, 4348833380280444, 13265036911604648, 40502401300634184
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OFFSET
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1,5
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COMMENTS
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Compare to: C(x) = x + C( C(x)^2 - C(x)^4 ) holds when C(x) = x + C(x)^2 is a g.f. of the Catalan numbers (A000108).
Compare to: F(x) = x + F( F(x)^3 - F(x)^9 ) holds when F(x) = x + F(x)^3 is a g.f. of the ternary tree numbers (A001764).
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LINKS
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FORMULA
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G.f. satisfies:
(1) A(x - A(x^3 + x^4)) = x.
(2) A(x) = x + Sum_{n>=0} d^n/dx^n A(x^3+x^4)^(n+1) / (n+1)!.
(3) A(x) = x * exp( Sum_{n>=0} d^n/dx^n A(x^3+x^4)^(n+1)/x / (n+1)! ).
(4) A(x) = x + G(x) + G(G(x)) + G(G(G(x))) + G(G(G(G(x)))) + ... where G(x) = A(x)^3 + A(x)^4.
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EXAMPLE
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G.f.: A(x) = x + x^3 + x^4 + 3*x^5 + 7*x^6 + 16*x^7 + 45*x^8 + 111*x^9 + 311*x^10 + 834*x^11 + 2329*x^12 + 6521*x^13 + 18429*x^14 + 52667*x^15 + ...
such that A(x) = x + A( A(x)^3 + A(x)^4 ).
RELATED SERIES.
A(x)^2 = x^2 + 2*x^4 + 2*x^5 + 7*x^6 + 16*x^7 + 39*x^8 + 110*x^9 + 277*x^10 + 786*x^11 + 2125*x^12 + 5996*x^13 + 16884*x^14 + 48044*x^15 + ...
A(x)^3 = x^3 + 3*x^5 + 3*x^6 + 12*x^7 + 27*x^8 + 70*x^9 + 198*x^10 + 510*x^11 + 1465*x^12 + 3999*x^13 + 11406*x^14 + 32328*x^15 + 92685*x^16 + ...
A(x)^4 = x^4 + 4*x^6 + 4*x^7 + 18*x^8 + 40*x^9 + 110*x^10 + 312*x^11 + 823*x^12 + 2392*x^13 + 6600*x^14 + 19032*x^15 + 54331*x^16 + ...
A(x)^3 + A(x)^4 = x^3 + x^4 + 3*x^5 + 7*x^6 + 16*x^7 + 45*x^8 + 110*x^9 + 308*x^10 + 822*x^11 + 2288*x^12 + 6391*x^13 + 18006*x^14 + 51360*x^15 + ...
A(x^3 + x^4) = x^3 + x^4 + x^9 + 3*x^10 + 3*x^11 + 2*x^12 + 4*x^13 + 6*x^14 + 7*x^15 + 16*x^16 + 30*x^17 + 37*x^18 + 57*x^19 + 108*x^20 + ...
where Series_Reversion(A(x)) = x - A(x^3 + x^4).
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PROG
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(PARI) {a(n) = my(A=x); for(i=1, n, A = x + subst(A, x, A^3 + A^4 +x*O(x^n))); polcoeff(H=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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