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%I #11 Jan 05 2014 03:10:03
%S 1,1,1,3,7,15,39,103,267,719,1975,5447,15199,42863,121647,347455,
%T 998559,2884143,8367599,24377503,71282351,209132511,615447711,
%U 1816255583,5373748287,15937008575,47368376255,141075930495,420957812863,1258317356799,3767538459391
%N G.f. satisfies: A(x) = (1 + x*(1-x)*A(x)) * (1 + x^2*A(x)^2).
%F G.f. satisfies: A(x) = exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n,k)^2 * x^k*(1-x)^(n-k) * A(x)^k ).
%F Recurrence: 2*(n+1)*(2*n+3)*(1367*n^5 - 22152*n^4 + 133091*n^3 - 342822*n^2 + 320852*n - 46056)*a(n) = (25973*n^7 - 386713*n^6 + 1995969*n^5 - 3489635*n^4 - 1128062*n^3 + 4916268*n^2 + 111144*n - 1105344)*a(n-1) - (42377*n^7 - 677143*n^6 + 3973061*n^5 - 9668137*n^4 + 6180338*n^3 + 7421072*n^2 - 9286128*n + 2230560)*a(n-2) + 4*(20505*n^7 - 365088*n^6 + 2506402*n^5 - 7967959*n^4 + 10412513*n^3 + 122671*n^2 - 10870392*n + 6017148)*a(n-3) - 2*(76552*n^7 - 1463333*n^6 + 10992560*n^5 - 39638307*n^4 + 64155864*n^3 - 18295816*n^2 - 55893000*n + 38191680)*a(n-4) + 2*(76552*n^7 - 1553555*n^6 + 12502721*n^5 - 49103379*n^4 + 89371647*n^3 - 37863250*n^2 - 72758304*n + 53344368)*a(n-5) - 4*(20505*n^7 - 443007*n^6 + 3803465*n^5 - 16027031*n^4 + 31715194*n^3 - 16244518*n^2 - 25294728*n + 19078920)*a(n-6) + 4*(n-6)*(6835*n^6 - 116228*n^5 + 733337*n^4 - 1976556*n^3 + 1537404*n^2 + 1530848*n - 1351680)*a(n-7) - 4*(n-7)*(n-6)*(1367*n^5 - 15317*n^4 + 58153*n^3 - 62791*n^2 - 47292*n + 44280)*a(n-8). - _Vaclav Kotesovec_, Dec 21 2013
%F a(n) ~ c*d^n/n^(3/2), where d = 3.14415377058430689... is the root of the equation -4 + 16*d - 44*d^2 + 68*d^3 - 44*d^4 + 16*d^5 - 15*d^6 + 4*d^7 = 0 and c = 0.77951549908443860621183... - _Vaclav Kotesovec_, Dec 21 2013
%F Constant c = 1/(8*sqrt(Pi*r)), where r = 0.008185036943737927662526644... is the root of the equation -5468 + 436073*r + 8837888*r^2 + 2268581888*r^3 - 4115660800*r^4 + 2178940928000*r^5 + 3543348019200*r^6 + 1717986918400*r^7 = 0. - _Vaclav Kotesovec_, Jan 04 2014
%e G.f.: A(x) = 1 + x + x^2 + 3*x^3 + 7*x^4 + 15*x^5 + 39*x^6 + 103*x^7 +...
%e The logarithm of the g.f. begins:
%e log(A(x)) = ((1-x) + x*A(x))*x + ((1-x)^2 + 2^2*x*(1-x)*A(x) + x^2*A(x)^2)*x^2/2 +
%e ((1-x)^3 + 3^2*x*(1-x)^2*A(x) + 3^2*x^2*(1-x)*A(x)^2 + x^3*A(x)^3)*x^3/3 +
%e ((1-x)^4 + 4^2*x*(1-x)^3*A(x) + 6^2*x^2*(1-x)^2*A(x)^2 + 4^2*x^3*(1-x)*A(x)^3 + x^4*A(x)^4)*x^4/4 +
%e ((1-x)^5 + 5^2*x*(1-x)^4*A(x) + 10^2*x^2*(1-x)^3*A(x)^2 + 10^2*x^3*(1-x)^2*A(x)^3 + 5^2*x^4*(1-x)*A(x)^4 + x^5*A(x)^5)*x^5/5 +...
%e Explicitly,
%e log(A(x)) = x + x^2/2 + 7*x^3/3 + 17*x^4/4 + 41*x^5/5 + 133*x^6/6 + 393*x^7/7 + 1121*x^8/8 + 3373*x^9/9 + 10161*x^10/10 +...
%o (PARI) {a(n)=local(A=1+x);for(i=1,n,A=(1 + x*(1-x)*A)*(1+x^2*A^2) +x*O(x^n));polcoeff(A,n)}
%o (PARI) {a(n)=local(A=1+x);for(i=1,n,A=exp(sum(m=1,n+1,x^m/m*sum(k=0,m,binomial(m,k)^2*x^k*(1-x)^(m-k)*A^k +x*O(x^n)))));polcoeff(A,n)}
%o for(n=0,40,print1(a(n),", "))
%Y Cf. A216617, A216604, A216447.
%K nonn
%O 0,4
%A _Paul D. Hanna_, Sep 10 2012
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