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A194786
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E.g.f. satisfies: A(x) = 1 + x*A(x)^A(x).
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3
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1, 1, 2, 12, 108, 1360, 21780, 424998, 9774912, 259012080, 7769656800, 260283596760, 9631680917760, 390185658289128, 17175153440774784, 816267894739416000, 41658264473400852480, 2272233977181361580160, 131913883517800157429760, 8121310193676734923381056
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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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E.g.f.: 1 + Series_Reversion( x * Sum_{n>=0} (-x)^n/n! * Product_{k=1..n} (k+x) ). - Paul D. Hanna, Sep 27 2014
a(n) = n!*Sum_{k=0..n-1} ((Sum_{i=k..n-1} (Stirling1(i,k)*binomial(k, n-i-1)/i!))*n^(k-1)), n > 0, a(0)=1. - Vladimir Kruchinin, Jan 24 2012
a(n) ~ n^(n-1) * (s-1)*sqrt(s/(1+(s-1)*s)) / (exp(n)*r^n), where s = 1.662886128060660201... is the root of the equation (s-1)*(1+log(s)) = 1, and r = (s-1)/s^s = 0.2845572964785024040... . - Vaclav Kotesovec, Jul 15 2014
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EXAMPLE
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E.g.f.: A(x) = 1 + x + 2*x^2/2! + 12*x^3/3! + 108*x^4/4! + 1360*x^5/5! + ...
where A(x)^A(x) = 1 + x + 4*x^2/2! + 27*x^3/3! + 272*x^4/4! + 3630*x^5/5! + ...
The e.g.f. also satisfies:
(1/x)*Series_Reversion(A(x) - 1) = 1 - x*(1+x) + x^2*(1+x)*(2+x)/2! - x^3*(1+x)*(2+x)*(3+x)/3! + x^4*(1+x)*(2+x)*(3+x)*(4+x)/4! - x^5*(1+x)*(2+x)*(3+x)*(4+x)*(5+x)/5! +- ...
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MATHEMATICA
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Flatten[{1, Table[n!*Sum[Sum[StirlingS1[i, k]*Binomial[k, n-i-1]/i!*n^(k-1), {i, k, n-1}], {k, 0, n-1}], {n, 1, 20}]}] (* Vaclav Kotesovec, Jul 15 2014 after Vladimir Kruchinin *)
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PROG
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(PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+x*(A+x*O(x^n))^A); n!*polcoeff(A, n)}
(Maxima) a(n):=if n=0 then 1 else (n!*sum((sum((stirling1(i, k)*binomial(k, n-i-1))/i!, i, k, n-1))*n^(k-1), k, 0, n-1)); /* Vladimir Kruchinin, Jan 24 2012 */
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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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