A060691 -id:A060691 - cp's OEIS frontend

A171454 G.f. satisfies: A(x) = 1 + 4xAGM(1, A(x)^2).

1, 4, 16, 80, 448, 2672, 16640, 106944, 704000, 4722608, 32166784, 221865280, 1546491904, 10876777024, 77091573760, 550088739584, 3948410757120, 28489277352112, 206520803651712, 1503353875355200, 10984898330047488, 80540719266134080, 592362120108263424, 4369140213882013440

Offset: 0

Paul D. Hanna, Dec 09 2009

nonn

			G.f.: A(x) = 1 + 4*x + 16*x^2 + 80*x^3 + 448*x^4 + 2672*x^5 + ...
A(x)^2 = 1 + 8*x + 48*x^2 + 288*x^3 + 1792*x^4 + 11488*x^5 + ...
AGM(1, A(x)^2) = 1 + 4*x + 20*x^2 + 112*x^3 + 668*x^4 + 4160*x^5 + ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..400

Cf. A060691.

(* Calculation of constants {d,c}: *) {1/r, Sqrt[r*s*(-1 - s + s^4 + s^5) / (2*Pi*r*(-1 - 4*s^2 + s^4) + 8*s*EllipticK[(-1 + s^2)^2/(1 + s^2)^2])]} /. FindRoot[{(s - 1)/(4*r) == Pi*s^2/(2*EllipticK[1 - 1/s^4]), EllipticE[(-1 + s^2)^2/(1 + s^2)^2] == Pi*r*s}, {r, 1/8}, {s, 3}, WorkingPrecision -> 70] (* Vaclav Kotesovec, Nov 15 2023 *)

{a(n)=local(A=1+x+x*O(x^n));for(i=1,n,A=1+4*x*agm(1,A^2));polcoeff(A,n)}

a(n) ~ c * d^n / n^(3/2), where d = 7.862810359254200633908490328120234046255594283562932892563... and c = 1.2926544621133576475917023125188188972684483736846308027... - Vaclav Kotesovec, Nov 15 2023

More terms from Jinyuan Wang, Feb 25 2020

A158100 G.f. satisfies: A(x) = 1/AGM(1, 1 - 8*x/A(x) ).

Original entry on oeis.org

1, 4, 4, 0, 4, 0, -16, 0, -28, 0, 176, 0, 336, 0, -2496, 0, -4956, 0, 40112, 0, 81488, 0, -694720, 0, -1432688, 0, 12647488, 0, 26360896, 0, -238598400, 0, -501256668, 0, 4623092400, 0, 9772018896, 0, -91458048960, 0, -194263943664, 0, 1839634167360

Offset: 0

Paul D. Hanna, Mar 13 2009

sign

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

			G.f.: A(x) = 1 + 4*x + 4*x^2 + 4*x^4 - 16*x^6 - 28*x^8 +-...
1 - 8*x/A(x) = 1 - 8*x + 32*x^2 - 96*x^3 + 256*x^4 - 608*x^5 +-...
From _Paul D. Hanna_, Mar 14 2009: (Start)
Convolution square root is A158122 and begins:
[1,2,0,0,2,-4,0,0,-16,40,0,0,200,-544,0,0,-3006,8540,0,0,...]
in which the convolution of the quadrisections equals 2:
[1,2,-16,200,-3006,...]*[2,-4,40,-544,8540,...] = 2. (End)

Cf. A060691, A158101 (bisection), A258053.

Cf. A158122 (sqrt), A158212, A158213.

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

G.f.: A(x) = x/Series_Reversion( x/AGM(1, 1-8*x) ).

Convolution square-root is A158122, which has two nonzero quadrisections, A158212 and A158213, that are inverse convolutions of each other (by a factor of 2). - Paul D. Hanna, Mar 14 2009

A158122 G.f. A(x) satisfies: A(x)^2 = 1/AGM(1, 1 - 8*x/A(x)^2 ).

Original entry on oeis.org

1, 2, 0, 0, 2, -4, 0, 0, -16, 40, 0, 0, 200, -544, 0, 0, -3006, 8540, 0, 0, 49956, -145720, 0, 0, -884352, 2625648, 0, 0, 16349648, -49161024, 0, 0, -311986480, 947069352, 0, 0, 6098614912, -18650752400, 0, 0, -121497078016, 373773754912, 0, 0

Offset: 0

Paul D. Hanna, Mar 13 2009

sign

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

			G.f.: A(x) = 1 + 2*x + 2*x^4 - 4*x^5 - 16*x^8 + 40*x^9 + 200*x^12 -+...
A(x)^2 = 1 + 4*x + 4*x^2 + 4*x^4 - 16*x^6 - 28*x^8 + 176*x^10 +...
Contribution from _Paul D. Hanna_, Mar 14 2009: (Start)
G.f. of quadrasection A158212 is:
B(x) = 1 + 2*x - 16*x^2 + 200*x^3 - 3006*x^4 + 49956*x^5 +...;
G.f. of quadrasection A158213 is C(x) = 2/B(x):
C(x) = 2 - 4*x + 40*x^2 - 544*x^3 + 8540*x^4 - 145720*x^5 +...
where g.f. A(x) = B(x^4) + x*C(x^4) = B(x^4) + 2*x/B(x^4). (End)

Cf. A060691, A158100 (self-convolution), A258053.

Cf. quadrasections: A158212, A158213.

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

G.f.: A(x) = sqrt( x/Series_Reversion( x/AGM(1,1-8*x) ) ).
Self-convolution equals A158100.
Contribution from Paul D. Hanna, Mar 14 2009: (Start)
Quadrasections are A158212(n) = A158122(4n) and A158213 = A158122(4n+1);
let B(x), C(x), be the g.f.s of A158212 and A158213, respectively,
then C(x) = 2/B(x) so that
A(x) = B(x^4) + x*C(x^4) = B(x^4) + 2*x/B(x^4) = 2/C(x^4) + x*C(x^4). (End)

A158212 A quadrisection of A158122: a(n) = A158122(4n).

Original entry on oeis.org

1, 2, -16, 200, -3006, 49956, -884352, 16349648, -311986480, 6098614912, -121497078016, 2457844837376, -50353474318552, 1042571366405520, -21781950163497216, 458626034728146240, -9721961867347898174

Offset: 0

Paul D. Hanna, Mar 14 2009

sign

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

			G.f.: A(x) = 1 + 2*x - 16*x^2 + 200*x^3 - 3006*x^4 + 49956*x^5 +...
2/A(x) = 2 - 4*x + 40*x^2 - 544*x^3 + 8540*x^4 - 145720*x^5 +...
F(x) = 1 + 2*x + 2*x^4 - 4*x^5 - 16*x^8 + 40*x^9 + 200*x^12 - 544*x^13 +...

Paul D. Hanna, Table of n, a(n) for n=0..100

Cf. A158122, A158213, A158100, A060691.

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

G.f.: A(x) = 2/B(x) where B(x) is the g.f. of A158213;
let F(x) = A(x^4) + 2*x/A(x^4) be the g.f. of A158122
then F(x) satisfies: F(x)^2 = 1/AGM(1, 1 - 8*x/F(x)^2 ).

A158213 A quadrisection of A158122: a(n) = A158122(4n+1).

Original entry on oeis.org

2, -4, 40, -544, 8540, -145720, 2625648, -49161024, 947069352, -18650752400, 373773754912, -7598155324032, 156294309718944, -3247203559571136, 68042170392274560, -1436308791802028544, 30514944039812500572

Offset: 0

Paul D. Hanna, Mar 14 2009

sign

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

			G.f.: A(x) = 2 - 4*x + 40*x^2 - 544*x^3 + 8540*x^4 - 145720*x^5 +...
2/A(x) = 1 + 2*x - 16*x^2 + 200*x^3 - 3006*x^4 + 49956*x^5 +...
F(x) = 1 + 2*x + 2*x^4 - 4*x^5 - 16*x^8 + 40*x^9 + 200*x^12 - 544*x^13 +...
where F(x) = 2/A(x^4) + x*A(x^4) is the g.f. of A158122.

Cf. A158122, A158212, A158100, A060691.

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

G.f.: A(x) = 2/B(x) where B(x) is the g.f. of A158212;
let F(x) = 2/A(x^4) + x*A(x^4) be the g.f. of A158122
then F(x) satisfies: F(x)^2 = 1/AGM(1, 1 - 8*x/F(x)^2 ).

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

Original entry on oeis.org

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

Paul D. Hanna, Mar 13 2009

sign

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

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

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.

Cf. A060691, A158100.
Cf. A002894. - Paul D. Hanna, Feb 04 2010
Cf. A003496.

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

{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

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)

A323385 Expansion of AGM(1,1-16*x) (where AGM denotes the arithmetic-geometric mean).

Original entry on oeis.org

1, -8, -16, -128, -1344, -15872, -199680, -2613248, -35148800, -482500608, -6730072064, -95094702080, -1358152794112, -19573573681152, -284284750397440, -4156674357067776, -61133523873169408, -903754859816157184, -13421680957337894912, -200140704802846801920

Offset: 0

Seiichi Manyama, Jan 13 2019

sign

Robert Israel, Table of n, a(n) for n = 0..833
Eric Weisstein's World of Mathematics, Arithmetic-geometric mean
Wikipedia, Arithmetic-geometric mean

Convolution inverse of A053175.

Cf. A060691, A089602, A090004.

R:= Pi*(1-8*x)/(2*EllipticK(-8*x/(1-8*x))):
S:= series(R,x,31):
seq(coeff(S,x,j),j=0..30); # Robert Israel, Jan 13 2019

CoefficientList[Series[Pi*(1 - 16*x) / (2*EllipticK[1 - 1/(1 - 16*x)^2]), {x, 0, 25}], x] (* Vaclav Kotesovec, Sep 28 2019 *)

PARI
```
N=66; x='x+O('x^N); Vec(agm(1, 1-16*x))
```

a(n) = 2^n * A060691(n).

a(n) ~ -Pi * 2^(4*n-1) / (n * log(n)^2) * (1 - (2*gamma + 4*log(2))/log(n) + (3*gamma^2 + 12*log(2)*gamma + 12*log(2)^2 - Pi^2/2) / log(n)^2), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Sep 29 2019

A328046 G.f.: 1/2 + 1/(1 + AGM(1, sqrt(1-16*x))).

Original entry on oeis.org

1, 1, 7, 68, 763, 9276, 118656, 1572024, 21368155, 296187164, 4169180104, 59420124472, 855590919392, 12425933510200, 181787367119112, 2676258927443328, 39615617922076635, 589234154312057436, 8801406013366190952, 131964659304934491576, 1985338775295068132520

Offset: 0

Vaclav Kotesovec, Oct 03 2019

nonn

AGM(x,y) = AGM((x+y)/2,sqrt(x*y)) is the arithmetic-geometric mean.

Vaclav Kotesovec, Table of n, a(n) for n = 0..830
Eric Weisstein's World of Mathematics, Arithmetic-Geometric Mean

Cf. A188267.

Cf. A060691, A081085, A323385.

CoefficientList[Series[1/2 + 1/(1 + (Pi*Sqrt[1 - 16*x])/(2*EllipticK[1 - 1/(1 - 16*x)])), {x, 0, 25}], x]

a(n) ~ Pi * 16^n / (n * (log(n) + Pi)^2) * (1 - (2*gamma + 8*log(2)) / (log(n) + Pi) + (3*gamma^2 + 48*log(2)^2 + 24*gamma*log(2) - Pi^2/2) / (log(n) + Pi)^2), where gamma is the Euler-Mascheroni constant A001620.

A300100 Expansion of sqrt(agm(1, 1 - 8*x)) in powers of x.

Original entry on oeis.org

1, -2, -4, -16, -82, -476, -2968, -19360, -130220, -895592, -6264656, -44411968, -318300080, -2302042400, -16777460032, -123084642048, -908175062994, -6734680013532, -50164119638328, -375134475461088, -2815268948389212, -21195313970398536

Offset: 0

Seiichi Manyama, Feb 24 2018

sign

Seiichi Manyama, Table of n, a(n) for n = 0..500
Eric Weisstein's World of Mathematics, Arithmetic-geometric mean
Wikipedia, Arithmetic-geometric mean

Cf. A060691, A089603.

CoefficientList[Series[Sqrt[Pi*(1 - 8*x) / (2*EllipticK[1 - 1/(1 - 8*x)^2])], {x, 0, 25}], x] (* Vaclav Kotesovec, Sep 28 2019 *)

Convolution inverse of A089603.

a(n) ~ -sqrt(Pi) * 2^(3*n - 3/2) / (n * log(n)^(3/2)) * (1 - 3*(gamma/2 + log(2)) / log(n) + (15*gamma^2/8 + 15*log(2)*gamma/2 + 15*log(2)^2/2 - 5*Pi^2/16) / log(n)^2), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Sep 29 2019

A075225 Expansion of 2-AGM(1,1-8x) (where AGM denotes the arithmetic-geometric mean).

Original entry on oeis.org

1, 4, 4, 16, 84, 496, 3120, 20416, 137300, 942384, 6572336, 46432960, 331580272, 2389352256, 17351364160, 126851634432, 932823545428, 6895102385072, 51199649648048, 381738099675840, 2856639909232112, 21447771308542784

Offset: 0

Michael Somos, Sep 11 2002

nonn

Cf. A060691. a(n)=-A060691(n) if n>0.

CoefficientList[Series[2 - Pi*(1 - 8*x) / (2*EllipticK[1 - 1/(1 - 8*x)^2]), {x, 0, 25}], x] (* Vaclav Kotesovec, Sep 28 2019 *)

a(n)=if(n<0,0,polcoeff(2-agm(1,1-8*x+x*O(x^n)),n))

G.f.: 2-AGM(1, 1-8x).

a(n) ~ Pi * 2^(3*n-1) / (n * log(n)^2) * (1 - (2*gamma + 4*log(2))/log(n) + (3*gamma^2 + 12*log(2)*gamma + 12*log(2)^2 - Pi^2/2) / log(n)^2), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Sep 29 2019

cp's OEIS Frontend

A171454 G.f. satisfies: A(x) = 1 + 4*x*AGM(1, A(x)^2).

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A158100 G.f. satisfies: A(x) = 1/AGM(1, 1 - 8*x/A(x) ).

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A158122 G.f. A(x) satisfies: A(x)^2 = 1/AGM(1, 1 - 8*x/A(x)^2 ).

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A158212 A quadrisection of A158122: a(n) = A158122(4n).

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A158213 A quadrisection of A158122: a(n) = A158122(4n+1).

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

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A323385 Expansion of AGM(1,1-16*x) (where AGM denotes the arithmetic-geometric mean).

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A328046 G.f.: 1/2 + 1/(1 + AGM(1, sqrt(1-16*x))).

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A171454 G.f. satisfies: A(x) = 1 + 4xAGM(1, A(x)^2).

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