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A161968 E.g.f. L(x) satisfies: L(x) = x*exp(x*d/dx L(x)), where L(x) is the logarithm of e.g.f. of A161967.

%I A161968 #12 Feb 25 2015 16:59:26
%S A161968 1,2,15,232,5905,220176,11210479,743759360,62179950753,6387468716800,
%T A161968 790466735915791,115974842104378368,19906425428056709425,
%U A161968 3952505003715017695232,899034956269244372091375,232282033898506324396343296,67660142460130946247667502401
%N A161968 E.g.f. L(x) satisfies: L(x) = x*exp(x*d/dx L(x)), where L(x) is the logarithm of e.g.f. of A161967.
%F A161968 a(n) = n * A156326(n-1), where the e.g.f. of A156326 satisfies: Sum_{n>=0} A156326(n)*x^n/n!  =  exp( Sum_{n>=1} n^2 * A156326(n-1)*x^n/n! )  =  exp( Sum_{n>=1} n * a(n)*x^n/n! ). - _Paul D. Hanna_, Feb 21 2014
%F A161968 E.g.f. A(x), with offset=0, satisfies [_Paul D. Hanna_, Feb 15 2015]:
%F A161968 (1) A(x) = d/dx x*exp(x*A(x)).
%F A161968 (2) A(x) = exp(x*A(x)) * (1 + x*A(x) + x^2*A'(x)).
%F A161968 (3) exp(x*A(x)) = e.g.f. of A156326.
%e A161968 E.g.f.: L(x) = x + 2*x^2/2! + 15*x^3/3! + 232*x^4/4! + 5905*x^5/5! +...
%e A161968 where exp(L(x)) = exp(x*exp(x*L'(x))) = e.g.f. of A161967:
%e A161968 exp(L(x)) = 1 + x + 3*x^2/2! + 22*x^3/3! + 317*x^4/4! + 7596*x^5/5! +...
%e A161968 and exp(x*L'(x)) = 1 + x + 5*x^2/2! + 58*x^3/3! + 1181*x^4/4! + 36696*x^5/5! +...+ A156326(n)*x^n/n! +...
%e A161968 RELATED EXPRESSIONS.
%e A161968 E.g.f.: A(x) = 1 + 2*x + 15*x^2/2! + 232*x^3/3! + 5905*x^4/4! +...
%e A161968 where
%e A161968 A(x) = d/dx x*exp(x*A(x)) = exp(x*A(x)) * (1 + x*A(x) + x^2*A'(x)) with
%e A161968 exp(x*A(x)) = 1 + x + 5*x^2/2! + 58*x^3/3! + 1181*x^4/4! + 36696*x^5/5! + 1601497*x^6/6! + 92969920*x^7/7! +...+ A156326(n)*x^n/n! +...
%o A161968 (PARI) {a(n)=local(L=x+x^2);for(i=1,n,L=x*exp(x*deriv(L)+O(x^n)));n!*polcoeff(L,n)}
%o A161968 for(n=1,30,print1(a(n),", "))
%Y A161968 Cf. A161967 (exp), A156326.
%K A161968 nonn
%O A161968 1,2
%A A161968 _Paul D. Hanna_, Jun 23 2009