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 A192666 E.g.f. satisfies: A(x) = exp(-1)*Sum_{n>=0} (1 + x*A(x))^(n^2)/n!. 0

%I #18 Mar 30 2012 18:37:27

%S 1,2,21,444,14415,637268,35822203,2450234160,197807272289,

%T 18431380399184,1948783220129813,230702141895062720,

%U 30251527782113610991,4355262112839582661824,683368350046603022039867,116136704024677305164141056

%N E.g.f. satisfies: A(x) = exp(-1)*Sum_{n>=0} (1 + x*A(x))^(n^2)/n!.

%C Compare to e.g.f. W(x) = LambertW(-x)/(-x) of A000272 (with offset) generated by: W(x) = exp(-1)*Sum_{n>=0} (1+x*W(x))^n/n! = Sum_{n>=0} (n+1)^(n-1)*x^n/n!.

%F E.g.f.: A(x) = Series_Reversion(x/G(x))/x, where G(x) = A(x/G(x)) = e.g.f. of A014507.

%e E.g.f.: A(x) = 1 + 2*x + 21*x^2/2! + 444*x^3/3! + 14415*x^4/4! +...

%e where A(x) = G(x*A(x)) and A(x/G(x)) = G(x) = e.g.f. of A014507:

%e G(x) = 1 + 2*x + 13*x^2/2! + 162*x^3/3! + 3075*x^4/4! + 80978*x^5/5! +...

%o (PARI) /* A(x) = 1/e*Sum_{n>=0}(1+x*A(x))^(n^2)/n! (requires precision): */

%o {a(n)=local(A=1+x);for(i=1,n,A=exp(-1)*sum(m=0,3*n+10,(1+x*A +x*O(x^n))^(m^2)/m!));polcoeff(round(serlaplace(A+x*O(x^n))),n)}

%o (PARI) /* E.g.f. Series_Reversion(x/G(x))/x; G(x) = e.g.f. of A014507: */

%o {Stirling1(n, k)=n!*polcoeff(binomial(x, n), k)}

%o {Bell(n)=n!*polcoeff(exp(exp(x+x*O(x^n))-1), n)}

%o {A014507(n)=sum(k=0, n, Stirling1(n, k)*Bell(2*k))}

%o {a(n)=local(G=sum(m=0,n,A014507(m)*x^m/m!)+x*O(x^n));n!*polcoeff(serreverse(x/G)/x,n)}

%Y Cf. A014507.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Jul 14 2011

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