The On-Line Encyclopedia of Integer Sequences (OEIS)

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A201465

E.g.f. satisfies: A(x) = (x + 2*exp(A(x)) - 2)/3.
(history; published version)

#24 by Paul D. Hanna at Fri Nov 21 19:25:06 EST 2014

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#23 by Paul D. Hanna at Fri Nov 21 19:25:02 EST 2014

PROG

for(n=0, 25, print1(round(A[n+1]), ", "))

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#22 by Paul D. Hanna at Mon Oct 27 22:53:57 EDT 2014

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approved

#21 by Paul D. Hanna at Mon Oct 27 22:53:44 EDT 2014

FORMULA

O.g.f.: Sum_{n>=0} 2^n / Product_{k=0..n} (3 - k*x). - Paul D. Hanna, Oct 27 2014

EXAMPLE

...

O.g.f.: G(x) = 1 + 2*x + 14*x^2 + 162*x^3 + 2622*x^4 + 54546*x^5 +...

where

G(x) = 1/3 + 2/(3*(3-x)) + 2^2/(3*(3-x)*(3-2*x)) + 2^3/(3*(3-x)*(3-2*x)*(3-3*x)) + 2^4/(3*(3-x)*(3-2*x)*(3-3*x)*(3-4*x)) + 2^5/(3*(3-x)*(3-2*x)*(3-3*x)*(3-4*x)*(3-5*x)) +...

PROG

(PARI) \p100 \\ set precision

{A=Vec(sum(n=0, 600, 1.*2^n/prod(k=0, n, 3 - k*x + O(x^31))))}

for(n=0, 25, print1(round(A[n+1]), ", ")) \\ Paul D. Hanna, Oct 27 2014

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editing

#20 by Bruno Berselli at Thu Dec 26 14:51:24 EST 2013

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#19 by Vaclav Kotesovec at Thu Dec 26 14:21:17 EST 2013

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#18 by Vaclav Kotesovec at Thu Dec 26 14:18:37 EST 2013

FORMULA

E.g.f.: (x-2)/3 - LambertW(-2/3*exp((x-2)/3)). - Vaclav Kotesovec, Dec 26 2013

a(n) ~ n^(n-1) / (sqrt(3) * exp(n) * (3*log(3)-3*log(2)-1)^(n-1/2)). - Vaclav Kotesovec, Dec 26 2013

MATHEMATICA

Rest[CoefficientList[1 + InverseSeries[Series[2 + 3*x - 2*Exp[x], {x, 0, 20}], x], x]* Range[0, 20]!] (* Vaclav Kotesovec, Dec 26 2013 *)

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#17 by Joerg Arndt at Wed May 01 13:09:13 EDT 2013

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#16 by Sergei N. Gladkovskii at Wed May 01 11:15:08 EDT 2013

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#15 by Sergei N. Gladkovskii at Wed May 01 11:14:57 EDT 2013

FORMULA

G.f.: 1/Q(0), where Q(k)= 1 - k*x - 2*x*(k+1)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 01 2013

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approved

editing