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%I #18 Jul 28 2021 22:38:33
%S 1,1,-1,-4,7,33,-68,-344,767,4035,-9425,-50832,122436,671804,-1653776,
%T -9189488,22992655,129001239,-326863667,-1847900500,4729547023,
%U 26903463697,-69424933968,-396930961632,1031309398852,5921685690388,-15474833826028,-89179284390112,234201961398776,1353916407418200
%N G.f. satisfies: A(x) = 1/(1 - x*A(-x)^2).
%C Compare to: G(x) = 1/(1 - x*G(x)^2) where G(x) = 1 + x*G(x)^3.
%C Compare to: G(x) = 1/(1 - x*G(-x)) where G(x) = 1 + x*C(-x^2) and C(x) = 1 + x*C(x)^2.
%H Vaclav Kotesovec, <a href="/A243863/b243863.txt">Table of n, a(n) for n = 0..440</a>
%F G.f. satisfies: x*(A(x) + A(-x))^3 - x*(A(x) + A(-x))^2 = 2*(A(x) - A(-x)).
%F From _Vaclav Kotesovec_, Jun 15 2014: (Start)
%F G.f. satisfies: (1-x)*y - 2*x*y^2 + 2*x*y^3 - x^2*y^4 + x^2*y^5 = 1, where y=A(x).
%F Recurrence: 64*(n-1)*n*(2*n-1)*(2*n+1)*(97344*n^7 - 1687296*n^6 + 12307620*n^5 - 48939592*n^4 + 114477342*n^3 - 157378523*n^2 + 117615351*n - 36821682)*a(n) = -48*(n-1)*(2*n-1)*(259584*n^7 - 4255264*n^6 + 28913612*n^5 - 105040520*n^4 + 218720756*n^3 - 257978161*n^2 + 156355143*n - 36051210)*a(n-1) - 12*(35043840*n^11 - 747601920*n^10 + 7046084448*n^9 - 38652188640*n^8 + 136786604012*n^7 - 326872151500*n^6 + 535977209166*n^5 - 599646116255*n^4 + 445036801288*n^3 - 206251878585*n^2 + 52769481426*n - 5487093360)*a(n-2) + 12*(n-3)*(1460160*n^8 - 21131136*n^7 + 128186214*n^6 - 422759974*n^5 + 821575770*n^4 - 951902899*n^3 + 631809561*n^2 - 214667176*n + 27671400)*a(n-3) + 3*(n-4)*(n-3)*(3*n-11)*(3*n-10)*(97344*n^7 - 1005888*n^6 + 4228068*n^5 - 9303892*n^4 + 11456294*n^3 - 7773065*n^2 + 2627695*n - 329436)*a(n-4).
%F a(n) ~ sqrt(1+sqrt(3)) * (3*(45+26*sqrt(3)))^(n/2) * (cos(Pi*n/2) + sqrt(1+2/sqrt(3))*sin(Pi*n/2)) / (sqrt(Pi) * n^(3/2) * 2^(2*n+2)).
%F (End)
%F a(n) = Sum_{k=0..n} (-1)^k * A278881(n,k) for n>=0. - _Paul D. Hanna_, Dec 01 2016
%F a(n) = Sum_{k=0..n-1} (-1)^k * (n+k)!*(2*n-k-1)!/(k!*(n-k)!*(2*k+1)!*(2*n-2*k-1)!) for n>0 with a(0)=1. - _Paul D. Hanna_, Dec 15 2016
%F a(0) = 1; a(n) = Sum_{i=0..n-1} Sum_{j=0..n-i-1} (-1)^(n-i-1) * a(i) * a(j) * a(n-i-j-1). - _Ilya Gutkovskiy_, Jul 28 2021
%e G.f.: A(x) = 1 + x - x^2 - 4*x^3 + 7*x^4 + 33*x^5 - 68*x^6 - 344*x^7 + 767*x^8 + 4035*x^9 - 9425*x^10 - 50832*x^11 + 122436*x^12 +...
%e where
%e A(x)^2 = 1 + 2*x - x^2 - 10*x^3 + 7*x^4 + 88*x^5 - 68*x^6 - 946*x^7 + 767*x^8 + 11298*x^9 - 9425*x^10 - 144024*x^11 + 122436*x^12 +...
%e 1/A(x) = 1 - x + 2*x^2 + x^3 - 10*x^4 - 7*x^5 + 88*x^6 + 68*x^7 - 946*x^8 - 767*x^9 + 11298*x^10 + 9425*x^11 - 144024*x^12 +...
%o (PARI) {a(n)=local(A=1+x);for(i=1,n,A=1/((1-x*subst(A^2,x,-x +x*O(x^n)))));polcoeff(A,n)}
%o for(n=0,30,print1(a(n),", "))
%o (PARI) {a(n) = if(n==0,1, sum(k=0,n-1,(-1)^k* (n+k)!*(2*n-k-1)!/(k!*(n-k)!*(2*k+1)!*(2*n-2*k-1)!)))}
%o for(n=0,30,print1(a(n),", ")) \\ _Paul D. Hanna_, Dec 15 2016
%Y Cf. A278745, A278881, A278882.
%K sign
%O 0,4
%A _Paul D. Hanna_, Jun 12 2014
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