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A235372
E.g.f. satisfies: A'(x) = A(x)^6 * A(-x)^2 with A(0) = 1.
6
1, 1, 4, 40, 472, 8656, 170944, 4546240, 126145792, 4391814016, 157103133184, 6757912560640, 296079903219712, 15160207270014976, 786265905880858624, 46689395543841955840, 2797780732283446853632, 188996669833290940481536, 12849664459334761078718464
OFFSET
0,3
LINKS
FORMULA
E.g.f.: 1/(1 - 3*Series_Reversion( Integral (1 - 9*x^2)^(2/3) dx ))^(1/3).
Limit n->infinity (a(n)/n!)^(1/n) = 21 * GAMMA(7/6) / (2 * sqrt(Pi) * GAMMA(2/3)) = 4.05858343437... - Vaclav Kotesovec, Jan 29 2014
EXAMPLE
E.g.f.: A(x) = 1 + x + 4*x^2/2! + 40*x^3/3! + 472*x^4/4! + 8656*x^5/5! +...
Related series.
A(x)^6 = 1 + 6*x + 54*x^2/2! + 720*x^3/3! + 12312*x^4/4! + 262656*x^5/5! +...
A(-x)^2 = 1 - 2*x + 10*x^2/2! - 104*x^3/3! + 1360*x^4/4! - 25232*x^5/5! +-...
where A(x)^6*A(-x)^2 = A'(x).
Note that 1 - 1/A(x)^3 is an odd function:
1 - 1/A(x)^3 = 3*x + 36*x^3/3! + 4968*x^5/5! + 1964736*x^7/7! +...
where Series_Reversion((1 - 1/A(x)^3)/3) = Integral (1-9*x^2)^(2/3) dx.
MATHEMATICA
CoefficientList[1/(1 - 3*InverseSeries[Series[Integrate[(1 - 9*x^2)^(2/3), x], {x, 0, 20}], x])^(1/3), x] * Range[0, 20]! (* Vaclav Kotesovec, Jan 28 2014 *)
PROG
(PARI) {a(n)=local(A=1); for(i=0, n, A=1+intformal(A^6*subst(A, x, -x)^2 +x*O(x^n) )); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n)=local(A=1); A=1/(1-3*serreverse(intformal((1-9*x^2 +x*O(x^n))^(2/3))))^(1/3); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jan 07 2014
STATUS
approved