A233895 - OEIS

%I #14 Mar 18 2024 06:09:46

%S 1,1,2,3,10,18,60,115,410,822,2996,6174,22980,48324,182328,389187,

%T 1484410,3205710,12329988,26876586,104080812,228606012,890262984,

%U 1967830254,7699472676,17110322908,67215426440,150058534620,591517612616,1325828841480,5241992235888

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

%H Vaclav Kotesovec, <a href="/A233895/b233895.txt">Table of n, a(n) for n = 0..100</a>

%F Recurrence: n*(n+1)*(n+2)*(81*n^6 - 756*n^5 + 2565*n^4 - 3630*n^3 + 1150*n^2 + 1998*n - 1648)*a(n) =  - 6*n*(n+1)*(81*n^5 - 1026*n^4 + 4851*n^3 - 10458*n^2 + 10280*n - 3872)*a(n-1) + 12*n*(486*n^8 - 4536*n^7 + 14634*n^6 - 14085*n^5 - 23214*n^4 + 69355*n^3 - 66518*n^2 + 29510*n - 6544)*a(n-2) - 72*(81*n^7 - 297*n^6 - 2124*n^5 + 13899*n^4 - 28141*n^3 + 20794*n^2 + 540*n - 4800)*a(n-3) - 144*(n-3)*(243*n^8 - 2268*n^7 + 7182*n^6 - 5976*n^5 - 13488*n^4 + 33301*n^3 - 30472*n^2 + 20528*n - 10400)*a(n-4) - 864*(n-4)*(n-3)*(n-2)*(108*n^3 - 441*n^2 + 75*n + 200)*a(n-5) - 1728*(n-5)*(n-4)*(n-3)*(81*n^6 - 270*n^5 + 690*n^3 - 695*n^2 + 374*n - 240)*a(n-6). - _Vaclav Kotesovec_, Dec 21 2013

%F a(n) ~ c*d^n/n^(3/2), where d = sqrt(24 - 3*I*2^(2/3)*3^(5/6)*(3 + I*sqrt(3))^(1/3) + 6*I*2^(1/3)*3^(1/6)*(3 + I*sqrt(3))^(2/3) - 3*2^(2/3)*(9 + 3*I*sqrt(3))^(1/3)) = 3.12769717670219... is the root of the equation 1728 + 432*d^2 - 72*d^4 + d^6 = 0 and c = sqrt((34 - 4*sqrt(247) * sin(arccsc(494 * sqrt(247)/7687)/3)) / Pi) = 1.281119572461999772722... if n is even, and c = 2*sqrt(6 - sqrt(129) * sin(arcsin(323*sqrt(3/43)/86)/3)) / sqrt(Pi) = 0.970593260725094233562... if n is odd. - _Vaclav Kotesovec_, Dec 21 2013, updated Mar 18 2024

%e G.f.: A(x) = 1 + x + 2*x^2 + 3*x^3 + 10*x^4 + 18*x^5 + 60*x^6 + 115*x^7 +...

%e Related series:

%e A(x)^2 = 1 + 2*x + 5*x^2 + 10*x^3 + 30*x^4 + 68*x^5 + 205*x^6 + 482*x^7 +...

%e A(x)*A(-x) = 1 + 3*x^2 + 18*x^4 + 115*x^6 + 822*x^8 + 6174*x^10 +...

%e A(x)^2+A(-x)^2 = 2 + 10*x^2 + 60*x^4 + 410*x^6 + 2996*x^8 + 22980*x^10 +...

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

%o for(n=0, 60, print1(a(n), ", "))

%Y Cf. A208887, A143926, A233896.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Dec 17 2013