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 A231552 G.f. satisfies: A(x) = (1 + x*A(x))^2 * (1 + 2*A(x)) / 3. 9
 1, 6, 63, 822, 12000, 187632, 3073047, 52042038, 903885102, 16012472484, 288207995934, 5255620271028, 96890977167126, 1802878079461764, 33814809038629719, 638631994011883878, 12134650158220342566, 231809353013520774756, 4449450392012510721474, 85770509189659517857524 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 LINKS G. C. Greubel, Table of n, a(n) for n = 0..750 FORMULA G.f. A(x) satisfies: (1) A(x) = exp( x*(A(x) + 2*A(x)^2) + Integral(A(x) + 2*A(x)^2 dx) ). (2) A(x) = (1/x)*Series_Reversion( x*(1-4*x-2*x^2)/(1+x)^2 ). (3) A(x) = 1 + x*A(x)*(1 + 2*A(x))*(2 + x*A(x)). (4) A(x) = 1 + Sum_{n>=2} (-1)^n * 3*n * x^(n-1) * A(x)^n. Recurrence: 8*n*(n+1)*(19*n-25)*a(n) = 4*n*(798*n^2 - 1449*n + 529)*a(n-1) - (513*n^3 - 1701*n^2 + 1630*n - 432)*a(n-2) + 2*(n-3)*(2*n-3)*(19*n-6)*a(n-3). - Vaclav Kotesovec, Dec 29 2013 a(n) ~ 2^(2*n-5/2) / (n^(3/2) * sqrt(Pi*s) * r^n), where r = 0.1919459762582734141... is the root of the equation -128 + 672*r - 27*r^2 + r^3 = 0 and s = 0.0229965315195763941... is the root of the equation -19 + 798*s + 1215*s^2 + 512*s^3 = 0. - Vaclav Kotesovec, Dec 29 2013 EXAMPLE G.f.: A(x) = 1 + 6*x + 63*x^2 + 822*x^3 + 12000*x^4 + 187632*x^5 + ... Related expansions: (1 + x*A(x))^2 = 1 + 2*x + 13*x^2 + 138*x^3 + 1806*x^4 + 26400*x^5 + ... (1 + 2*A(x))/3 = 1 + 4*x + 42*x^2 + 548*x^3 + 8000*x^4 + 125088*x^5 + ... A(x) + 2*A(x)^2 = 3 + 30*x + 387*x^2 + 5622*x^3 + 87666*x^4 + 1433304*x^5 + ... log(A(x)) = 6*x + 90*x^2/2 + 1548*x^3/3 + 28110*x^4/4 + 525996*x^5/5 + ... MATHEMATICA CoefficientList[1/x*InverseSeries[Series[x*(1-4*x-2*x^2)/(1+x)^2, {x, 0, 20}], x], x] (* Vaclav Kotesovec, Dec 29 2013 *) PROG (PARI) {a(n)=polcoeff((serreverse(x*(1-4*x-2*x^2)/(1+x)^2 +x^2*O(x^n))/x), n)} for(n=0, 20, print1(a(n), ", ")) (PARI) {a(n)=local(A=1); for(i=1, n, A=exp(x*(A+2*A^2)+intformal(A+2*A^2 +x*O(x^n)))); polcoeff(A, n)} for(n=0, 20, print1(a(n), ", ")) (PARI) {a(n)=local(A=1); for(i=1, n, A=1+x*A*(1+2*A)*(2+x*A) +x*O(x^n)); polcoeff(A, n)} for(n=0, 20, print1(a(n), ", ")) CROSSREFS Cf. A228966, A231553, A231554, A231556, A231615, A231616, A231618. Sequence in context: A210987 A341376 A234465 * A302103 A229451 A132078 Adjacent sequences:  A231549 A231550 A231551 * A231553 A231554 A231555 KEYWORD nonn AUTHOR Paul D. Hanna, Nov 10 2013 STATUS approved

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Last modified August 1 22:36 EDT 2021. Contains 346408 sequences. (Running on oeis4.)