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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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..365

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

Cf. A235370, A235371, A235373, A235374.

Sequence in context: A264112 A304447 A002705 * A034385 A249927 A218302

Adjacent sequences:  A235369 A235370 A235371 * A235373 A235374 A235375

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jan 07 2014

STATUS

approved

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Last modified May 17 23:07 EDT 2022. Contains 353779 sequences. (Running on oeis4.)