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A171190 G.f. satisfies: A(x) = exp( Sum_{n>=1} (A(x)^n + A(-x)^n) * x^n/n ).

%I A171190 #14 Feb 12 2024 02:26:25
%S A171190 1,2,3,10,27,112,336,1490,4791,22138,74079,351288,1207620,5831208,
%T A171190 20436516,100004994,355610367,1758044950,6322608561,31511387450,
%U A171190 114359284515,573713781760,2097612975456,10580600244664,38925304968612
%N A171190 G.f. satisfies: A(x) = exp( Sum_{n>=1} (A(x)^n + A(-x)^n) * x^n/n ).
%H A171190 Paul D. Hanna, <a href="/A171190/b171190.txt">Table of n, a(n) for n = 0..300</a>
%F A171190 G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
%F A171190 (1) A(x) = exp( Sum_{n>=1} (A(x)^n + A(-x)^n) * x^n/n ).
%F A171190 (2) A(x) = 1/((1 - x*A(x)) * (1 - x*A(-x))). - _Paul D. Hanna_, Dec 06 2009
%F A171190 (3) 0 = 1 - (3-x)*A(x) + (2-x)*A(x)^2 - (2-5*x)*x*A(x)^3 - (2+x)*x^2*A(x)^4 + 2*x^3*A(x)^5. - _Paul D. Hanna_, Feb 11 2024
%e A171190 G.f.: A(x) = 1 + 2*x + 3*x^2 + 10*x^3 + 27*x^4 + 112*x^5 + 336*x^6 + 1490*x^7 + 4791*x^8 + 22138*x^9 + 74079*x^10 + 351288*x^11 + 1207620*x^12 + ...
%e A171190 where the logarithm of A(x) may be written as
%e A171190 log(A(x)) = (A(x) + A(-x))*x + (A(x)^2 + A(-x)^2)*x^2/2 + (A(x)^3 + A(-x)^3)*x^3/3 + (A(x)^4 + A(-x)^4)*x^4/4 + (A(x)^5 + A(-x)^5)*x^5/5 + ...
%e A171190 Incidentally, the square root of g.f. A(x) is an integer series starting
%e A171190 A(x)^(1/2) = 1 + x + x^2 + 4*x^3 + 9*x^4 + 43*x^5 + 108*x^6 + 558*x^7 + 1517*x^8 + 8175*x^9 + 23219*x^10 + 128516*x^11 + 375896*x^12 + ...
%o A171190 (PARI) {a(n) = my(A=1+x+x*O(x^n)); for(i=1,n, A=exp(sum(m=1,n,(A^m+subst(A^m,x,-x)+x*O(x^n))*x^m/m)));polcoeff(A,n)}
%o A171190 (PARI) {a(n) = my(A=1+x); for(i=1,n, A=(1-x*A+x*O(x^n))^-1*(1-x*subst(A,x,-x)+x*O(x^n))^-1);polcoeff(A,n)} \\ _Paul D. Hanna_, Dec 06 2009
%Y A171190 Cf. A171191, A171199.
%K A171190 nonn
%O A171190 0,2
%A A171190 _Paul D. Hanna_, Dec 05 2009