A158101 G.f. satisfies: A(x^2) = -4*x + 1/AGM(1, 1 - 8*x/(A(x^2) + 4*x) ).

1, 4, 4, -16, -28, 176, 336, -2496, -4956, 40112, 81488, -694720, -1432688, 12647488, 26360896, -238598400, -501256668, 4623092400, 9772018896, -91458048960, -194263943664, 1839634167360, 3923099632704, -37510172125440

Offset: 0

Author

Paul D. Hanna, Mar 13 2009

Keywords

• sign

Comments

See A060691 for the expansion of AGM(1,1-8x), where AGM denotes the arithmetic-geometric mean.

Examples

			G.f.: A(x) = 1 + 4*x + 4*x^2 - 16*x^3 - 28*x^4 + 176*x^5 + 336*x^6 - ...
		

Links

• Otto Dirk, Störungstheorie des Anderson-Modells: Untersuchung und Erweiterung der NCA und DMFT, Dr. rer. nat. thesis, Universität Dortmund, 2003 [in German]. It appears that the table on p. 48 contains this sequence.

Crossrefs

Cf. A060691, A158100.

Cf. A002894. - Paul D. Hanna, Feb 04 2010

Cf. A003496.

Programs

PARI

{a(n)=polcoeff(-4*x+x/serreverse(x/agm(1, 1-8*x +O(x^(2*n+1)))),2*n)}

PARI

{a(n)=local(G=sum(m=0,n,binomial(2*m,m)^2*x^m)+x*O(x^n));polcoeff((x/serreverse(x*G^2))^(1/2),n)} \\ Paul D. Hanna, Feb 04 2010

Formula

A bisection of A158100.

G.f. satisfies: A(x^2) = -4*x + x/Series_Reversion( x/AGM(1,1-8*x) ).

From Paul D. Hanna, Feb 04 2010: (Start)

G.f. satisfies: A(x) = Sum_{n>=0} C(2n,n)^2*x^n/A(x)^(2n).

G.f.: A(x) = [x/Series_Reversion(x*G(x)^2)]^(1/2) where G(x) = Sum_{n>=0} C(2n,n)^2*x^n = 1/AGM(1, (1-16*x)^(1/2)) = g.f. of A002894.

(End)