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 1-3 of 3 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

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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

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

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 + 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)

Crossrefs

Cf. A060691, A158100 (self-convolution), A258053.
Cf. quadrasections: A158212, A158213.

Programs

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

Formula

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

Views

Author

Paul D. Hanna, Mar 14 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 + 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 +...

Links

Crossrefs

Cf. A158122, A158213, A158100, A060691.

Programs

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

Formula

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 ).
Showing 1-3 of 3 results.

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