A245931 -id:A245931 - cp's OEIS frontend

A245932 G.f.: G'(x) / G(x) where G(x) = 1 / sqrt( AGM((1 - 3*x)^2, (1 + x)^2) ) is the g.f. of A245931.

1, 1, 1, 9, 41, 121, 281, 673, 2017, 6721, 21121, 61065, 171497, 495769, 1488761, 4509793, 13468897, 39688609, 116869153, 346788009, 1035199817, 3090560089, 9200749433, 27347417281, 81352371841, 242426988961, 723125351521, 2156829477609, 6430792717001, 19174372701241, 57194628447641

Offset: 0

Paul D. Hanna, Aug 14 2014

nonn

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

Limit a(n+1)/a(n) = 3.

Paul D. Hanna, Table of n, a(n) for n = 0..1000

Cf. A245931.

Simplify[CoefficientList[Series[D[Log[Sqrt[(2*EllipticK[1 - (1 - 3*x)^4/(1 + x)^4])/Pi] / (1 + x)], x], {x, 0, 30}], x]] (* Vaclav Kotesovec, Sep 27 2019 *)

/* As the logarithmic derivative of A245931: */
{a(n)=local(G=1); G = 1 / sqrt( agm((1-3*x)^2, (1+x)^2 +x^2*O(x^n)) ); polcoeff(G'/G,n)}
for(n=0,35,print1(a(n),", "))

/* As the logarithm of g.f. of A245931 (offset = 1): */
{a(n)=local(A=1); A = -log( agm((1-3*x)^2, (1+x)^2 +x*O(x^n)) )/2; n*polcoeff(A,n)}
for(n=1,35,print1(a(n),", "))

G.f.: A(x) = 1 + x + x^2 + 9*x^3 + 41*x^4 + 121*x^5 + 281*x^6 + 673*x^7 +...
As a logarithmic expansion,
L(x) = x + x^2/2 + x^3/3 + 9*x^4/4 + 41*x^5/5 + 121*x^6/6 + 281*x^7/7 + 673*x^8/8 + 2017*x^9/9 + 6721*x^10/10 +...
where
exp(L(x)) = 1 + x + x^2 + x^3 + 3*x^4 + 11*x^5 + 31*x^6 + 71*x^7 + 157*x^8 +...
equals 1 / sqrt( AGM((1 - 3*x)^2, (1 + x)^2) ).
a(n) ~ 3^(n+1) / (2*log(n)) * (1 + (log(3) - 3*log(2) - gamma) / log(n)), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Sep 30 2019

A245930 G.f.: 1 / AGM((1 - 3*x)^2, (1 + x)^2).

Original entry on oeis.org

1, 2, 3, 4, 9, 30, 91, 232, 549, 1378, 3839, 11100, 31301, 85694, 234207, 653328, 1856829, 5300010, 15062839, 42702596, 121448901, 347414166, 997886671, 2870139480, 8257776521, 23782773242, 68627659563, 198437633884, 574654851209, 1665825647430, 4833258038251, 14037680955552, 40816416373293

Offset: 0

Paul D. Hanna, Aug 14 2014

nonn

Here AGM(x,y) = AGM((x+y)/2, sqrt(x*y)) denotes the arithmetic-geometric mean.
Self-convolution of A245931.
Limit a(n+1)/a(n) = 3.

			G.f.: A(x) = 1 + 2*x + 3*x^2 + 4*x^3 + 9*x^4 + 30*x^5 + 91*x^6 + 232*x^7 +...
where
1/A(x) = 1 - 2*x + x^2 - 4*x^4 - 8*x^5 - 12*x^6 - 16*x^7 - 40*x^8 - 144*x^9 - 448*x^10 - 1152*x^11 - 2732*x^12 - 6840*x^13 - 18964*x^14 +...
equals AGM((1 - 3*x)^2, (1 + x)^2).
SPECIFIC VALUES:
A(x) = 2 at x = 0.2650276124990406644...
A(x) = 3 at x = 0.31872724866867463...
A(x) = 4 at x = 0.32990867978741...

Paul D. Hanna, Table of n, a(n) for n = 0..1000

Cf. A245931, A245932, A227845.

{a(n)=local(A=1); A = 1 / agm((1-3*x)^2, (1+x)^2 +x*O(x^n)); polcoeff(A,n)}
for(n=0,35,print1(a(n),", "))

G.f.: 1 / AGM((1-x)^2 + 4*x^2, (1-x)^2 - 4*x^2).
G.f.: 1 / AGM((1-x)^2, sqrt((1-x)^4 - 16*x^4)).
Recurrence: n^2*a(n) = (5*n^2 - 5*n + 2)*a(n-1) - 2*(5*n^2 - 10*n + 6)*a(n-2) + 2*(5*n^2 - 15*n + 12)*a(n-3) + 11*(n-2)^2*a(n-4) - 15*(n-3)*(n-2)*a(n-5). - Vaclav Kotesovec, Aug 16 2014
a(n) ~ 3^(n+2) / (4*Pi*n). - Vaclav Kotesovec, Aug 16 2014

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A245932 G.f.: G'(x) / G(x) where G(x) = 1 / sqrt( AGM((1 - 3*x)^2, (1 + x)^2) ) is the g.f. of A245931.

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A245930 G.f.: 1 / AGM((1 - 3*x)^2, (1 + x)^2).

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