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%I #7 Mar 25 2014 09:18:38
%S 1,1,2,11,38,257,1040,7646,33374,256718,1171454,9270560,43558064,
%T 351490167,1686018600,13799914556,67223728270,556203232266,
%U 2741975026412,22880729474777,113875773363274,956800135969601
%N G.f. satisfies: A(x) = 1 + x*A(x)^4*A(-x)^2.
%H Vaclav Kotesovec, <a href="/A143550/b143550.txt">Table of n, a(n) for n = 0..370</a>
%H Vaclav Kotesovec, <a href="/A143550/a143550.txt">Recurrence</a>
%F G.f. satisfies: A(x) + A(-x) = 1 + [A(x)*A(-x)] + x^2*[A(x)*A(-x)]^6.
%F G.f. satisfies: 1 - 4*y + 6*y^2 - 4*y^3 + y^4 - 2*x*y^6 + 4*x*y^7 - x*y^8 - x*y^9 + x^2*y^12 = 0, where y=A(x). - _Vaclav Kotesovec_, Mar 25 2014
%F a(n) ~ c / (sqrt(Pi)*n^(3/2)*r^n)), where r = sqrt(22444621 + 5142958*sqrt(19))/46656 = 0.143559867369277217..., c = sqrt((13 - 49/sqrt(19))/3)/3 = 0.255214437... if n is even, and c = sqrt((73 - 1/sqrt(19))/3)/15 = 0.328341701... if n is odd. - _Vaclav Kotesovec_, Mar 25 2014
%e G.f. A(x) = 1 + x + 2*x^2 + 11*x^3 + 38*x^4 + 257*x^5 + 1040*x^6 +...
%e Related expansions:
%e A(x)^4 = 1 + 4*x + 14*x^2 + 72*x^3 + 333*x^4 + 1936*x^5 + 9966*x^6 +...
%e A(-x)^2 = 1 - 2*x + 5*x^2 - 26*x^3 + 102*x^4 - 634*x^5 + 2867*x^6 -+...
%e A(x)^2*A(-x) = 1 + x + 5*x^2 + 14*x^3 + 102*x^4 + 348*x^5 + 2867*x^6 +...
%e A(x)*A(-x) = 1 + 3*x^2 + 58*x^4 + 1597*x^6 + 51406*x^8 + 1807747*x^10 +...
%e [A(x)*A(-x)]^6 = 1 + 18*x^2 + 483*x^4 + 15342*x^6 + 535161*x^8 +...
%o (PARI) {a(n)=local(A=1+x*O(x^n));for(i=0,2*n,A=1+x*A^4*subst(A^2,x,-x));polcoeff(A,n)}
%Y Cf. A143338, A143549, A143551, A143552, A143553, A143554.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Aug 24 2008
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