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A194471
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E.g.f. A(x) satisfies: A(x) = exp(x) + x*A(x)^2.
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4
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1, 2, 9, 79, 1065, 19401, 445933, 12389021, 403897553, 15120448273, 639345572181, 30138682861365, 1567316344601593, 89137628104427033, 5503952108613407933, 366697176991277153341, 26220726323043177903009, 2002962250253424509250081
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OFFSET
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0,2
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COMMENTS
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The radius of convergence r of the e.g.f. satisfies: r = exp(-r)/4 = limit (n+1)*a(n)/a(n+1) = 0.203888354702240... with A(r) = 1/(2*r) = 2.452322501352287...
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LINKS
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FORMULA
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E.g.f.: A(x) = (1 - sqrt(1 - 4*x*exp(x))) / (2*x).
a(n) = 1 + n*Sum_{k=0..n-1} C(n-1,k)*a(k)*a(n-1-k) for n>=0.
a(n) ~ sqrt(2)*sqrt(1+LambertW(1/4))*n^(n-1)/(4*exp(n)*LambertW(1/4)^(n+1)). - Vaclav Kotesovec, Aug 19 2013
a(n) = n!*sum(k=0..n, (k+1)^(n-k-1)*binomial(2*k,k)/(n-k)!). - Vladimir Kruchinin, Sep 01 2014
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EXAMPLE
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E.g.f.: A(x) = 1 + 2*x + 9*x^2/2! + 79*x^3/3! + 1065*x^4/4! +...
Related expansion:
A(x)^2 = 1 + 4*x + 26*x^2/2! + 266*x^3/3! + 3880*x^4/4! + 74322*x^5/5! +...
Illustrate the recurrence:
a(2) = 1 + 2*(1*1*2 + 1*2*1) = 1 + 2*4 = 9;
a(3) = 1 + 3*(1*1*9 + 2*2*2 + 1*9*1) = 1 + 3*26 = 79;
a(4) = 1 + 4*(1*1*79 + 3*2*9 + 3*9*2 + 1*79*1) = 1 + 4*266 = 1065;
a(5) = 1 + 5*(1*1*1065 + 4*2*79 + 6*9*9 + 4*79*2 + 1*1065*1) = 1 + 5*3880 = 19401.
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MATHEMATICA
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f[0] = 1; f[n_] := f[n] = 1 + n*Sum[ Binomial[n - 1, k]*f[k]*f[n - 1 - k] , {k, 0, n - 1}]; Array[f, 18, 0] (* Robert G. Wilson v, Aug 25 2011 *)
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PROG
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(PARI) {a(n)=n!*polcoeff((1 - sqrt(1 - 4*x*exp(x +O(x^(n+2))))) / (2*x), n)}
(PARI) {a(n)=1+n*sum(k=0, n-1, binomial(n-1, k)*a(k)*a(n-1-k))}
(Maxima) a(n):=n!*sum((k+1)^(n-k-1)*binomial(2*k, k)/(n-k)!, k, 0, n); /* Vladimir Kruchinin, Sep 01 2014 */
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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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