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A212722
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E.g.f. satisfies: A(x) = exp( x/(1 - x*A(x)^2) ).
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12
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1, 1, 3, 25, 313, 5341, 115651, 3036517, 93767185, 3330162073, 133737097411, 5992748728561, 296433923379529, 16044427276953973, 943207466055927619, 59848531677741706621, 4076826825898115406241, 296742863575079244130225
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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} n! * (1 + 2*(n-k))^(k-1)/k! * C(n-1,n-k).
Let A(x)^m = Sum_{n>=0} a(n,m)*x^n/n!, then
a(n,m) = Sum_{k=0..n} n! * m*(m + 2*(n-k))^(k-1)/k! * C(n-1,n-k).
a(n) ~ n^(n-1) * (1+1/(2*c))^(n+1/2) / (2*sqrt(1+c) * exp(n) * c^n), where c = LambertW(1/sqrt(2)) = 0.450600515864833072257... . - Vaclav Kotesovec, Jul 15 2014
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EXAMPLE
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E.g.f: A(x) = 1 + x + 3*x^2/2! + 25*x^3/3! + 313*x^4/4! + 5341*x^5/5! + ...
such that, by definition:
log(A(x))/x = 1 + x*A(x)^2 + x^2*A(x)^4 + x^3*A(x)^6 + x^4*A(x)^8 + ...
Related expansions:
log(A(x)) = x/(1-x*A(x)^2) = x + 2*x^2/2! + 18*x^3/3! + 216*x^4/4! + 3640*x^5/5! + 78000*x^6/6! + 2032464*x^7/7! + 62400128*x^8/8! + ... + n*A366232(n-1)*x^n/n! + ...
A(x)^2 = 1 + 2*x + 8*x^2/2! + 68*x^3/3! + 880*x^4/4! + 15312*x^5/5! + 336064*x^6/6! +...
A(x)^4 = 1 + 4*x + 24*x^2/2! + 232*x^3/3! + 3232*x^4/4! + 59104*x^5/5! + 1343296*x^6/6! +...
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MATHEMATICA
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Table[Sum[n! * (1 + 2*(n-k))^(k-1)/k! * Binomial[n-1, n-k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jul 15 2014 *)
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PROG
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(PARI) {a(n, m=1)=if(n==0, 1, sum(k=0, n, n!/k!*m*(m+2*(n-k))^(k-1)*binomial(n-1, n-k)))}
(PARI) {a(n, m=1)=local(A=1+x); for(i=1, n, A=exp(x/(1-x*A^2+x*O(x^n)))); n!*polcoeff(A^m, n)}
for(n=0, 21, 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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