A233895 G.f. satisfies: A(x) = 1 + x*A(x)*A(-x) + x^2*(A(x)^2 + A(-x)^2).

1, 1, 2, 3, 10, 18, 60, 115, 410, 822, 2996, 6174, 22980, 48324, 182328, 389187, 1484410, 3205710, 12329988, 26876586, 104080812, 228606012, 890262984, 1967830254, 7699472676, 17110322908, 67215426440, 150058534620, 591517612616, 1325828841480, 5241992235888

Offset: 0

Author

Paul D. Hanna, Dec 17 2013

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 3*x^3 + 10*x^4 + 18*x^5 + 60*x^6 + 115*x^7 +...
Related series:
A(x)^2 = 1 + 2*x + 5*x^2 + 10*x^3 + 30*x^4 + 68*x^5 + 205*x^6 + 482*x^7 +...
A(x)*A(-x) = 1 + 3*x^2 + 18*x^4 + 115*x^6 + 822*x^8 + 6174*x^10 +...
A(x)^2+A(-x)^2 = 2 + 10*x^2 + 60*x^4 + 410*x^6 + 2996*x^8 + 22980*x^10 +...
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..100

Crossrefs

Cf. A208887, A143926, A233896.

Programs

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)}
for(n=0, 60, print1(a(n), ", "))

Formula

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

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