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A231618
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G.f. satisfies: A(x) = (1 + 3*x*A(x))^2 * (3 + A(x)) / 4.
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8
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1, 8, 92, 1264, 19220, 311736, 5285868, 92578656, 1661895332, 30416141032, 565437707516, 10647518728016, 202669742580916, 3893175999963032, 75376952791465356, 1469420261169552576, 28817874206182559556, 568178789630200669128, 11255461001317809903516
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OFFSET
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0,2
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LINKS
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FORMULA
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G.f. A(x) satisfies:
(1) A(x) = exp( x*(3*A(x) + A(x)^2) + Integral(3*A(x) + A(x)^2 dx) ).
(2) A(x) = (1/x)*Series_Reversion( x*(1+x)*(1-3*x)/(1+3*x)^2 ).
(3) A(x) = 1 + x*A(x)*(3 + A(x))*(2 + 3*x*A(x)).
(4) A(x) = 1 + Sum_{n>=2} (-3)^(n-2) * 4*n * x^(n-1) * A(x)^n.
Recurrence: 2*n*(n+1)*(31*n-45)*a(n) = n*(1426*n^2 - 2783*n + 1143)*a(n-1) - 18*(2*n-1)*(62*n^2 - 183*n + 126)*a(n-2) + 81*(n-3)*(2*n-3)*(31*n-14)*a(n-3). - Vaclav Kotesovec, Dec 29 2013
a(n) ~ 2^(n-1) * 3^(5*n-1) / (n^(3/2) * sqrt(Pi*s) * r^n), where r = 22.6037423012063424... is the root of the equation -1417176 + 67068*r - 216*r^2 + r^3 = 0 and s = 0.009747169749742737082... is the root of the equation -31 + 1116*s + 174960*s^2 + 3779136*s^3 = 0. - Vaclav Kotesovec, Dec 29 2013
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EXAMPLE
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G.f.: A(x) = 1 + 8*x + 92*x^2 + 1264*x^3 + 19220*x^4 + 311736*x^5 +...
Related expansions.
(1 + 3*x*A(x))^2 = 1 + 6*x + 57*x^2 + 696*x^3 + 9816*x^4 + 151320*x^5 +...
(3 + A(x))/4 = 1 + 2*x + 23*x^2 + 316*x^3 + 4805*x^4 + 77934*x^5 +...
3*A(x) + A(x)^2 = 4 + 40*x + 524*x^2 + 7792*x^3 + 124788*x^4 + 2098776*x^5 +...
log(A(x)) = 8*x + 120*x^2/2 + 2096*x^3/3 + 38960*x^4/4 + 748728*x^5/5 +...
The square root of the g.f. is an integer series:
sqrt(A(x)) = 1 + 4*x + 38*x^2 + 480*x^3 + 6968*x^4 + 109756*x^5 +...
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MATHEMATICA
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CoefficientList[1/x*InverseSeries[Series[x*(1+x)*(1-3*x)/(1+3*x)^2, {x, 0, 20}], x], x] (* Vaclav Kotesovec, Dec 29 2013 *)
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PROG
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(PARI) {a(n)=polcoeff((serreverse(x*(1+x)*(1-3*x)/(1+3*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*(3*A+A^2)+intformal(3*A+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*(3+A)*(2+3*x*A) +x*O(x^n)); polcoeff(A, n)}
for(n=0, 20, 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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