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 1-4 of 4 results.

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 ).

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

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) = 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.

Crossrefs

Cf. A158122, A158212, A158100, A060691.

Programs

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

Formula

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 ).

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

A258053 G.f. A(x) satisfies: 1 = Sum_{n>=0} x^n / (A(x)^2 - x)^(2*n+1) * [ Sum_{k=0..n} C(n,k)^2 * x^k * A(x)^(n-2*k) ]^2.

Original entry on oeis.org

1, 1, 1, 0, 1, -2, -4, 0, -7, 20, 42, 0, 84, -272, -584, 0, -1239, 4270, 9288, 0, 20370, -72860, -159840, 0, -358092, 1312824, 2897400, 0, 6587944, -24580512, -54490848, 0, -125256887, 473534676, 1053346410, 0, 2441648384, -9325376200, -20800103016, 0, -48534848222, 186886877456, 417769301220, 0
Offset: 0

Views

Author

Paul D. Hanna, May 17 2015

Keywords

Examples

			G.f.: A(x) = 1 + x + x^2 + x^4 - 2*x^5 - 4*x^6 - 7*x^8 + 20*x^9 + 42*x^10 + 84*x^12 - 272*x^13 - 584*x^14 - 1239*x^16 +...
RELATED SERIES.
A(x)^2 = 1 + 2*x + 3*x^2 + 2*x^3 + 3*x^4 - 2*x^5 - 10*x^6 - 12*x^7 - 21*x^8 + 22*x^9 + 106*x^10 +...
satisfies
A(x)^2 = G(x/A(x)^2) and A(x*G(x))^2 = G(x), where
G(x) = 1 + 2*x + 7*x^2 + 28*x^3 + 125*x^4 + 590*x^5 + 2891*x^6 + 14536*x^7 + 74497*x^8 +...+ A227845(n)*x^n +...
Also,
A(x) + x/A(x) = 1 + 2*x + 2*x^4 - 4*x^5 - 16*x^8 + 40*x^9 + 200*x^12 - 544*x^13 - 3006*x^16 +...+ A158122(n)*x^n +...
and
(A(x) + x/A(x))^2 = 1 + 4*x + 4*x^2 + 4*x^4 - 16*x^6 - 28*x^8 + 176*x^10 + 336*x^12 - 2496*x^14 +...+ A158100(n)*x^n +...
thus
A(x) = sqrt( (F-2*x + sqrt(F)*sqrt(F-4*x))/2 ) where F is the g.f. of A158100.

Crossrefs

Cf. A227845, A158122, A158100.

Programs

  • PARI
    {a(n)=local(A); A=sqrt(x/serreverse(x/agm((1+x)^2, 1-6*x+x^2 +x^2*O(x^n)))); polcoeff(A, n)}
for(n=0, 50, print1(a(n), ", "))

Formula

a(4*n+3) = 0 for n>=0.
G.f. A(x) satisfies:
(1) (A(x) + x/A(x))^2 = x / Series_Reversion( x/AGM(1,1-8*x) ),
(2) A(x) = sqrt( x / Series_Reversion(x*G(x)) ),
(3) A(x)^2 = G(x/A(x)^2),
(4) A(x*G(x))^2 = G(x),
where G(x) = 1 / AGM((1+x)^2, 1-6*x+x^2) is the g.f. of A227845, and AGM(x,y) = AGM((x+y)/2,sqrt(x*y)) denotes the arithmetic-geometric mean.
Showing 1-4 of 4 results.

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