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A161567
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E.g.f. satisfies: A(x) = exp(x*exp(x*A(x)^2)).
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5
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1, 1, 3, 22, 233, 3356, 61057, 1343686, 34731377, 1031493880, 34617603041, 1295705404874, 53516386593001, 2417918198462404, 118628419305036929, 6280926119941402486, 356960234149564116833, 21674784895404653181680
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OFFSET
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0,3
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LINKS
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FORMULA
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a(n) = Sum_{k=0..n} C(n,k) * (2*(n-k) + 1)^(k-1) * k^(n-k).
E.g.f.: A(x) = F(x)^(1/2) where F(x) = e.g.f. of A161566.
E.g.f.: A(x) = exp(x*G(x)) where G(x) = e.g.f. of A161565.
a(n) ~ n^(n-1) / (2*exp(n)*r^(n+1/2)), where r = 0.256263163133653382... is the root of the equation 1/LambertW(1/r) = -log(2*r^2) - LambertW(1/r). - Vaclav Kotesovec, Feb 28 2014
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EXAMPLE
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E.g.f.: A(x) = 1 + x + 3*x^2/2! + 22*x^3/3! + 233*x^4/4! +...
log(A(x)) = x*G(x) where G(x) = exp(x*A(x)^2) = e.g.f. of A161565:
G(x) = 1 + x + 5*x^2/2! + 37*x^3/3! + 417*x^4/4! + 6201*x^5/5! +...
A(x)^2 = 1 + 2*x + 8*x^2/2! + 62*x^3/3! + 696*x^4/4! + 10362*x^5/5! +...
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MATHEMATICA
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Flatten[{1, Table[Sum[Binomial[n, k] * (2*(n-k) + 1)^(k-1) * k^(n-k), {k, 0, n}], {n, 1, 20}]}] (* Vaclav Kotesovec, Feb 28 2014 *)
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PROG
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(PARI) {a(n)=sum(k=0, n, binomial(n, k)*(2*(n-k)+1)^(k-1)*k^(n-k))}
(PARI) {a(n)=local(A=1+x); for(i=0, n, A=exp(x*exp(x*A^2+O(x^n)))); n!*polcoeff(A, n, x)}
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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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