A158101 - OEIS

%I #10 Oct 19 2023 02:17:34

%S 1,4,4,-16,-28,176,336,-2496,-4956,40112,81488,-694720,-1432688,

%T 12647488,26360896,-238598400,-501256668,4623092400,9772018896,

%U -91458048960,-194263943664,1839634167360,3923099632704,-37510172125440

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

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

%H Otto Dirk, <a href="https://doi.org/10.17877/DE290R-14802">Störungstheorie des Anderson-Modells: Untersuchung und Erweiterung der NCA und DMFT</a>, Dr. rer. nat. thesis, Universität Dortmund, 2003 [in German]. It appears that the table on p. 48 contains this sequence.

%F A bisection of A158100.

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

%F From _Paul D. Hanna_, Feb 04 2010: (Start)

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

%F 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.

%F (End)

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

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

%o (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

%Y Cf. A060691, A158100.

%Y Cf. A002894. - _Paul D. Hanna_, Feb 04 2010

%Y Cf. A003496.

%K sign

%O 0,2

%A _Paul D. Hanna_, Mar 13 2009