cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-2 of 2 results.

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

Views

Author

Paul D. Hanna, Mar 13 2009

Keywords

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 + 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)

Crossrefs

Cf. A060691, A158101 (bisection), A258053.
Cf. A158122 (sqrt), A158212, A158213.

Programs

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

Formula

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

A172390 G.f. satisfies: A(x) = G(x/A(x))^2 and G(x)^2 = A(x*G(x)^2) where G(x) = Sum_{n>=0} C(2*n,n)^2*x^n.

Original entry on oeis.org

1, 8, 24, 0, -168, 0, 2112, 0, -32040, 0, 536256, 0, -9542976, 0, 177126912, 0, -3390361128, 0, 66436117440, 0, -1326185205696, 0, 26872637815296, 0, -551301904867392, 0, 11428295231789568, 0, -239010764560888320, 0
Offset: 0

Views

Author

Paul D. Hanna, Feb 04 2010

Keywords

Examples

			G.f.: A(x) = 1 + 8*x + 24*x^2 - 168*x^4 + 2112*x^6 - 32040*x^8 + ...
A(x) = G(x/A(x))^2 where G(x) = 1/AGM(1, (1-16*x)^(1/2)) is the power series:
G(x) = 1 + 2^2*x + 6^2*x^2 + 20^2*x^3 + 70^2*x^4 + 252^2*x^5 + ... + C(2*n,n)^2*x^n + ...
The square root of g.f. A(x) begins:
A(x)^(1/2) = 1 + 4*x + 4*x^2 - 16*x^3 - 28*x^4 + 176*x^5 + 336*x^6 + ... + A158101(n)*x^n + ...

Crossrefs

Cf. A036917, A002894, A158101, A158100.

Programs

  • 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),n)}
  • PARI
    {a(n)=if(n==1,8,polcoeff(agm(1,sqrt(1-16*x +x^2*O(x^n)))^(2*n-2),n)/(1-n))} \\ Paul D. Hanna, Mar 20 2010

Formula

G.f.: A(x) = x/Series_Reversion(x*G(x)^2) where G(x) = Sum_{n>=0} C(2*n,n)^2*x^n = 1/agm(1, (1-16*x)^(1/2)) = g.f. of A002894 and G(x)^2 is the g.f. of A036917.
Self-convolution of A158101, which is a bisection of A158100; A158100 has g.f. F(x) that satisfies: F(x) = 1/AGM(1, 1 - 8*x/F(x) ).
a(n) = [x^n] AGM(1,(1-16*x)^(1/2))^(2*n-2)/(1-n) for n>1 where AGM is the arithmetic-geometric mean of Gauss. - Paul D. Hanna, Mar 20 2010
Showing 1-2 of 2 results.

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