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A208888 G.f. satisfies: A(x) = 1 + x*(A(x) + A(-x)) + x^2*A(x)*A(-x). 2

%I #13 Mar 03 2016 11:09:49

%S 1,2,1,2,-2,-4,-11,-22,-14,-28,58,116,316,632,397,794,-2198,-4396,

%T -11954,-23908,-14684,-29368,95170,190340,517492,1034984,623764,

%U 1247528,-4462472,-8924944,-24270275,-48540550,-28820966,-57641932,220608454,441216908,1200216340

%N G.f. satisfies: A(x) = 1 + x*(A(x) + A(-x)) + x^2*A(x)*A(-x).

%H Paul D. Hanna, <a href="/A208888/b208888.txt">Table of n, a(n) for n = 0..300</a>

%F G.f.: A(x) = (1 - sqrt(1 - 4*x^2 + 16*x^4)) / (2*x^2*(1-2*x)).

%F Recurrence: (n+2)*a(n) = 2*(n+2)*a(n-1) + 4*(n-1)*a(n-2) - 8*(n-1)*a(n-3) - 16*(n-4)*a(n-4) + 32*(n-4)*a(n-5). - _Vaclav Kotesovec_, Aug 19 2013

%F |a(n)| ~ c * 3^(1/4)*2^(n+2)/(sqrt(Pi)*n^(3/2)), where c=(sqrt(3)+1)/2 if n=6k+0 or n=6k+1, c=(sqrt(3)-1)/2 if n=6k+2 or n=6k+3 and c=1 if n=6k+4 or n=6k+5. - _Vaclav Kotesovec_, Aug 19 2013

%e G.f.: A(x) = 1 + 2*x + x^2 + 2*x^3 - 2*x^4 - 4*x^5 - 11*x^6 - 22*x^7 +...

%e Related series:

%e A(x)+A(-x) = 2 + 2*x^2 - 4*x^4 - 22*x^6 - 28*x^8 + 116*x^10 + 632*x^12 +...

%e A(x)*A(-x) = 1 - 2*x^2 - 11*x^4 - 14*x^6 + 58*x^8 + 316*x^10 + 397*x^12 +...

%p A208888_list := proc(n) local j, a, w; a := array(0..n); a[0] := 1;

%p for w from 1 to n do a[w] := a[w-1] - add((-1)^j*a[j]*a[w-j-1], j=1..w-1) od;

%p convert(a, list); subsop(1=NULL,%); end: A208888_list(37); # _Peter Luschny_, Feb 29 2016

%t CoefficientList[Series[(1-Sqrt[1-4*x^2+16*x^4])/(2*x^2*(1-2*x)), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Aug 19 2013 *)

%o (PARI) {a(n)=local(A=1+x);for(i=1,n,A=1+x*(A+subst(A,x,-x+x*O(x^n)))+x^2*A*subst(A,x,-x+x*O(x^n)));polcoeff(A,n)}

%o for(n=0,40,print1(a(n),", "))

%Y Cf. A208887.

%K sign

%O 0,2

%A _Paul D. Hanna_, Mar 09 2012

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