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 A143556 G.f. satisfies: A(x) = 1 + x*A(x)^3/A(-x)^3. 6

%I #15 Mar 25 2014 11:52:01

%S 1,1,6,18,110,498,3366,17282,122958,672930,4938758,28103730,210595182,

%T 1230391058,9358456230,55727128866,428643977422,2589488117826,

%U 20092671283974,122759098980690,959216278565742,5913900861617970

%N G.f. satisfies: A(x) = 1 + x*A(x)^3/A(-x)^3.

%H Vaclav Kotesovec, <a href="/A143556/b143556.txt">Table of n, a(n) for n = 0..450</a>

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

%F G.f. satisfies: A(x) = 1 + x^2 + x*A(x)^3/A(-x)^2.

%F G.f. satisfies: (A(x) - 1)^2 = ( 1 - (1+x^2)/A(x) )^3/x = x^2*A(x)^6/A(-x)^6.

%F G.f.: A(x) = (1+x^2)*G(x) where G(x) = 1 + x*G(x)^3/G(-x)^2 is the g.f. of A143562.

%F G.f. satisfies: x*A(x)^5 - 2*x*A(x)^4 - (1-x)*A(x)^3 + 3*(1+x^2)*A(x)^2 - 3*(1+x^2)^2*A(x) + (1+x^2)^3 = 0.

%F Recurrence: 4*(n-1)*n*(2*n-5)*(2*n+1)*(2916*n^10 - 99630*n^9 + 1494855*n^8 - 12945798*n^7 + 71493183*n^6 - 262308129*n^5 + 645244282*n^4 - 1046448887*n^3 + 1066283852*n^2 - 614660500*n + 152638416)*a(n) = 60*(n-1)*(13122*n^11 - 458217*n^10 + 7044759*n^9 - 62741439*n^8 + 358008636*n^7 - 1365100815*n^6 + 3513825159*n^5 - 6010387373*n^4 + 6521940316*n^3 - 4078695988*n^2 + 1207261712*n - 113170176)*a(n-1) + 15*(n-2)*(160380*n^13 - 6121170*n^12 + 104460435*n^11 - 1051745310*n^10 + 6938544798*n^9 - 31476010053*n^8 + 100128993299*n^7 - 223244300184*n^6 + 341877397736*n^5 - 343306364591*n^4 + 206330136024*n^3 - 62025904772*n^2 + 8101283136*n - 2665897920)*a(n-2) + 450*(n-4)*(7020*n^10 - 107820*n^9 + 91377*n^8 + 9009842*n^7 - 87380558*n^6 + 404731832*n^5 - 1079876519*n^4 + 1690685386*n^3 - 1439622136*n^2 + 509372600*n + 4226320)*a(n-3) + 750*(n-5)*(n-4)*(14580*n^12 - 498150*n^11 + 7512345*n^10 - 65844630*n^9 + 371440818*n^8 - 1409248026*n^7 + 3643384398*n^6 - 6348642805*n^5 + 7178246227*n^4 - 4869145209*n^3 + 1716210104*n^2 - 292182404*n + 75613440)*a(n-4) + 3750*(n-6)*(n-5)*(n-4)*(3240*n^8 - 52785*n^7 + 324459*n^6 - 854916*n^5 + 387102*n^4 + 2695803*n^3 - 5239793*n^2 + 2713946*n + 268800)*a(n-5) + 3125*(n-7)*(n-6)*(n-5)*(n-4)*(2916*n^10 - 70470*n^9 + 729405*n^8 - 4223718*n^7 + 14971977*n^6 - 33317457*n^5 + 45697282*n^4 - 36099439*n^3 + 14258060*n^2 - 2573132*n + 694560)*a(n-6). - _Vaclav Kotesovec_, Mar 25 2014

%F a(n) ~ c / (sqrt(Pi)*n^(3/2)*r^n), where {r1 = r = 0.13384151194121538538097804723..., s1 = 1.57588974374012701113388456...} and {r2 = -r, s2 = 0.9688941320566492403600185...} are roots of the system of equations r*(r^5 + 3*r*(s-1)^2 + (s-1)^2*s^3) = 3*r^4*(s-1) + (s-1)^3, r*(s-1)*(6*r + s^2*(5*s-3)) = 3*(r^4 + (s-1)^2), and c = c1+c2 = 0.525673619703566161096484... if n is even, and c = c1-c2 = 0.471796676012154625609556... if n is odd, where c1 = M(r1,s1), c2=M(r2,s2), and M(r,s) = sqrt(r*(6*r^5 - 12*r^3*(s-1) + 6*r*(s-1)^2 + (s-1)^2*s^3)/(3+3*r^2-3*s+r*s*(3-12*s+10*s^2)))/2. - _Vaclav Kotesovec_, Mar 25 2014

%e G.f. A(x) = 1 + x + 6*x^2 + 18*x^3 + 110*x^4 + 498*x^5 + 3366*x^6 +...

%e A(x)/A(-x) = 1 + 2*x + 2*x^2 + 26*x^3 + 50*x^4 + 706*x^5 + 1650*x^6 +...

%e A(x)^2/A(-x)^2 = 1 + 4*x + 8*x^2 + 60*x^3 + 208*x^4 + 1716*x^5 +...

%e where 1 - (1+x^2)/A(x) = x*A(x)^2/A(-x)^2.

%o (PARI) {a(n)=local(A=1+x*O(x^n));for(i=0,n,A=1+x*A^3/subst(A^3,x,-x));polcoeff(A,n)}

%Y Cf. A143562, A143555, A143557, A143558, A143559.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Aug 24 2008

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