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A232692 E.g.f. satisfies: A(x) = exp( 1/A(x)^3 * Integral A(x)^8 dx ). 3
1, 1, 3, 24, 213, 3096, 46071, 967608, 20251809, 555747048, 15004870731, 508165972056, 16810393586733, 677183788645704, 26523956467895103, 1238567261126084856, 56056407696184372281, 2976966230117448265128, 152872356339113679491859, 9098430770913969095416728 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Compare e.g.f. to: B(x) = exp( 1/B(x)^3 * Integral B(x)^3 dx ) where B(y) = Bessel polynomial y_n(-3) (cf. A065923).

Note that G(x) = exp(1/G(x)^3 * Integral G(x)^7 dx) has negative coefficients.

CONJECTURE:

Given G(x,n,k) = G such that G = exp( 1/G^n * Integral G^k dx ) then G(x,n,k) consists solely of positive coefficients when k >= A047399(n) where A047399 lists numbers that are congruent to {0,3,6} mod 8.

LINKS

Paul D. Hanna, Table of n, a(n) for n = 0..300

FORMULA

E.g.f.: (3*LambertW(-1, (25*x-8)/3*exp(-8/3))/(25*x-8))^(1/5). - Vaclav Kotesovec, Jan 05 2014

EXAMPLE

E.g.f.: A(x) = 1 + x + 3*x^2/2! + 24*x^3/3! + 213*x^4/4! + 3096*x^5/5! +...

Related expansions:

log(A(x)) = x + 2*x^2/2! + 17*x^3/3! + 120*x^4/4! + 1905*x^5/5! + 23640*x^6/6! +...

Integral A(x)^8 dx = x + 8*x^2/2! + 80*x^3/3! + 1032*x^4/4! + 16320*x^5/5! +...

1/A(x)^3 = 1 - 3*x + 3*x^2/2! - 24*x^3/3! + 117*x^4/4! - 2088*x^5/5! +...

MAPLE

seq(n! * coeff(series((3*LambertW(-1, (25*x-8)/3*exp(-8/3))/(25*x-8))^(1/5), x, n+1), x, n), n=0..20) # Vaclav Kotesovec, Jan 05 2014

PROG

(PARI) {a(n)=local(A=1+x); for(i=1, n, A=exp(1/A^3*intformal(A^8+x*O(x^n)))); n!*polcoeff(A, n)}

for(n=0, 30, print1(a(n), ", "))

CROSSREFS

Cf. A232690, A232691.

Sequence in context: A308354 A073978 A278991 * A000279 A292311 A279973

Adjacent sequences:  A232689 A232690 A232691 * A232693 A232694 A232695

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Dec 06 2013

STATUS

approved

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Last modified October 22 18:08 EDT 2019. Contains 328319 sequences. (Running on oeis4.)