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

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

0,2

Comments

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

Links

Table of n, a(n) for n=0..43.

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)

Example

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)

Prog

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

Crossrefs

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

Cf. quadrasections: A158212, A158213.

Sequence in context: A020474 A135589 A244312 * A348165 A366531 A028641

Adjacent sequences: A158119 A158120 A158121 * A158123 A158124 A158125

Keyword

sign

Author

Paul D. Hanna, Mar 13 2009

Status

approved