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A136727
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E.g.f.: A(x) = [ exp(x)/(3 - 2*exp(x)) ]^(1/3).
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6
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1, 1, 3, 17, 139, 1481, 19443, 303297, 5480219, 112549881, 2589274883, 65957355377, 1842897053099, 56038776055081, 1842278768795923, 65109900167188257, 2461735422517374779, 99148196540813749081
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OFFSET
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0,3
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COMMENTS
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G.f. of variant A014307 is B(x) = sqrt(exp(x)/(2-exp(x))), which satisfies: B(x) = 1 + integral(B(x)^3*exp(-x)).
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LINKS
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FORMULA
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E.g.f. A(x) satisfies: A(x) = 1 + integral( A(x)^4 * exp(-x) ).
O.g.f.: 1/(1 - x/(1-2*x/(1 - 4*x/(1-4*x/(1 - 7*x/(1-6*x/(1 - 10*x/(1-8*x/(1 - 13*x/(1-10*x/(1 - ...)))))))))), a continued fraction.
G.f.: 1/G(0) where G(k) = 1 - x*(3*k+1)/( 1 - 2*x*(k+1)/G(k+1) ); (continued fraction ). - Sergei N. Gladkovskii, Mar 23 2013
a(n) ~ n! * sqrt(3)*2^(2/3)*Gamma(2/3)/(4*Pi*n^(2/3)*(log(3/2))^(n+1/3)). - Vaclav Kotesovec, Jun 25 2013
a(n) = 1 + 2 * Sum_{k=1..n-1} (binomial(n,k) - 1) * a(k). - Ilya Gutkovskiy, Jul 09 2020
a(n) = Sum_{k=0..n} (-1)^(n-k) * (Product_{j=0..k-1} (3*j+1)) * Stirling2(n,k).
a(0) = 1; a(n) = Sum_{k=1..n} (-1)^k * (2*k/n - 3) * binomial(n,k) * a(n-k).
a(0) = 1; a(n) = a(n-1) + 2*Sum_{k=1..n-1} binomial(n-1,k) * a(n-k). (End)
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EXAMPLE
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E.g.f.: A(x) = 1 + x + 3/2*x^2 + 17/6*x^3 + 139/24*x^4 + 1481/120*x^5 +...
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MATHEMATICA
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With[{nn=20}, CoefficientList[Series[(Exp[x]/(3-2Exp[x]))^(1/3), {x, 0, nn}], x] Range[0, nn]!] (* Harvey P. Dale, Jan 26 2013 *)
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PROG
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(PARI) {a(n) = n!*polcoeff((exp(x +x*O(x^n))/(3-2*exp(x +x*O(x^n))))^(1/3), n)}
for(n=0, 25, print1(a(n), ", "))
(PARI) /* As solution to integral equation: */
{a(n) = local(A=1+x+x*O(x^n)); for(i=0, n, A = 1 + intformal(A^4*exp(-x+x*O(x^n)))); n!*polcoeff(A, n)}
for(n=0, 25, 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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