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A200755
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G.f. satisfies A(x) = 1 + x*A(x)^3 - x^2*A(x)^2.
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3
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1, 1, 2, 7, 29, 129, 602, 2910, 14447, 73234, 377487, 1972568, 10425930, 55640282, 299403552, 1622701202, 8850030065, 48534971244, 267486182192, 1480673755443, 8228819436898, 45895682480965, 256815165790211, 1441321638029496, 8111194646903282
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OFFSET
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0,3
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COMMENTS
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Compare to the g.f. C(x) for the Catalan numbers (A000108): C(x) = 1 + x*C(x)^3 - x^2*C(x)^4 = 1 + x*C(x)^2.
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LINKS
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FORMULA
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Recurrence: 2*n*(2*n+1)*(244*n^3 - 1713*n^2 + 3767*n - 2550)*a(n) = 3*(2196*n^5 - 17613*n^4 + 49628*n^3 - 59841*n^2 + 30478*n - 5184)*a(n-1) - 18*(244*n^5 - 2323*n^4 + 8013*n^3 - 12252*n^2 + 7774*n - 1260)*a(n-2) - (n-4)*(244*n^4 - 1713*n^3 + 4172*n^2 - 3333*n - 378)*a(n-3) - 36*(n-5)*(n-3)*(5*n + 2)*a(n-4) - 4*(n-6)*(n-4)*(244*n^3 - 981*n^2 + 1073*n - 252)*a(n-5). - Vaclav Kotesovec, Sep 10 2013
a(n) ~ c*d^n/n^(3/2), where d = 5.991151107674316485... is the root of the equation -4 - 4*d - 5*d^2 - 23*d^3 + 4*d^4 = 0 and c = 0.214566307956522153666714736272121899143... - Vaclav Kotesovec, Sep 10 2013
a(n) = Sum_{k=0..floor(n/2)} (-1)^k * binomial(3*n-4*k,k) * binomial(3*n-5*k,n-2*k) / (2*n-3*k+1). - Seiichi Manyama, Nov 02 2023
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EXAMPLE
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G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 29*x^4 + 129*x^5 + 602*x^6 +...
Related expansions:
A(x)^2 = 1 + 2*x + 5*x^2 + 18*x^3 + 76*x^4 + 344*x^5 + 1627*x^6 +...
A(x)^3 = 1 + 3*x + 9*x^2 + 34*x^3 + 147*x^4 + 678*x^5 + 3254*x^6 +...
where a(2) = 3 - 1; a(3) = 9 - 2; a(4) = 34 - 5; a(5) = 147 - 18; ...
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MATHEMATICA
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nmax=20; aa=ConstantArray[0, nmax]; aa[[1]]=1; Do[AGF=1+Sum[aa[[n]]*x^n, {n, 1, j-1}]+koef*x^j; sol=Solve[Coefficient[1 + x*AGF^3 - x^2*AGF^2 - AGF, x, j]==0, koef][[1]]; aa[[j]]=koef/.sol[[1]], {j, 2, nmax}]; Flatten[{1, aa}] (* Vaclav Kotesovec, Sep 10 2013 *)
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PROG
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(PARI) a(n)=local(A=1+x); for(i=1, n, A=1+x*A^3-x^2*A^2+x*O(x^n)); polcoeff(A, n);
(PARI) a(n) = sum(k=0, n\2, (-1)^k*binomial(3*n-4*k, k)*binomial(3*n-5*k, n-2*k)/(2*n-3*k+1)); \\ Seiichi Manyama, Nov 02 2023
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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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