Logo

Greetings from The On-Line Encyclopedia of Integer Sequences!

Hints

Search: id:A104794
Displaying 1-1 of 1 results found. page 1
     Format: long | short | internal | text      Sort: relevance | references | number      Highlight: on | off
A104794 Expansion of theta_4(q)^2. +0
3
1, -4, 4, 0, 4, -8, 0, 0, 4, -4, 8, 0, 0, -8, 0, 0, 4, -8, 4, 0, 8, 0, 0, 0, 0, -12, 8, 0, 0, -8, 0, 0, 4, 0, 8, 0, 4, -8, 0, 0, 8, -8, 0, 0, 0, -8, 0, 0, 0, -4, 12, 0, 8, -8, 0, 0, 0, 0, 8, 0, 0, -8, 0, 0, 4, -16, 0, 0, 8, 0, 0, 0, 4, -8, 8, 0, 0, 0, 0, 0, 8, -4, 8, 0, 0, -16, 0, 0, 0, -8, 8, 0, 0, 0, 0, 0, 0, -8, 4, 0, 12 (list; graph; listen)
OFFSET

0,2

COMMENT

In the Arithmetic-Geometric Mean, if a=theta_3(q)^2, b=theta_4(q)^2 then a':=(a+b)/2=theta_3(q^2)^2, b':=sqrt(ab)=theta_4(q^2)^2.

REFERENCES

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 576.

J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987.

LINKS

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, December 1972 [alternative scanned copy].

FORMULA

Euler transform of period 2 sequence [ -4, -2, ...].

Expansion of eta(q)^4/eta(q^2)^2 in powers of q.

G.f. A(x) satisfies 0=f(A(x), A(x^2), A(x^4)) where f(u, v, w)=v(u^2+v^2)-2uw^2.

G.f.: theta_4(q)^2 = (Sum_{k} (-q)^(k^2))^2 = (Product_{k>0} (1-q^(2k))(1-q^(2k-1))^2)^2.

G.f. A(x) satisfies 0=f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6)=u1^2 -2*u1*u3 +4*u2*u6 -3*u3^2.

Moebius transform is period 8 sequence [ -4, 8, 4, 0, -4, -8, 4, 0, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 16 (t/i) g(t) where q = exp(2 pi i t) and g() is g.f. for A008441.

a(4*n+3) = 0.

EXAMPLE

1 - 4*q + 4*q^2 + 4*q^4 - 8*q^5 + 4*q^8 - 4*q^9 + 8*q^10 - 8*q^13 + ...

PROGRAM

(PARI) a(n)=if(n<1, n==0, (-1)^n*4*sumdiv(n, d, (d%4==1)-(d%4==3)))

(PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x+A)^4/eta(x^2+A)^2, n))}

(PARI) {a(n)=if(n<0, 0, polcoeff( 1+4*sum(k=1, n, (-x)^k/(1+x^(2*k)), x*O(x^n)), n))}

CROSSREFS

A004018(n) = (-1)^n * a(n) = a(2*n). -4 * A008441(n) = a(4*n+1). -4 * A113652(n) = a(n) unless n=0. 4 * A122865(n) = a(6*n+2). 4 * A122856(n) = a(6*n+4). -4 * A113407(n) = a(8*n+1). -8 * A053692(n) = a(8*n+5).

Sequence in context: A106508 A104287 A138518 this_sequence A004018 A028658 A028642

Adjacent sequences: A104791 A104792 A104793 this_sequence A104795 A104796 A104797

KEYWORD

sign

AUTHOR

Michael Somos, Mar 26 2005

page 1

Search completed in 0.002 seconds

Lookup | Welcome | Find friends | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Transforms | Puzzles | Hot | Classics
More pages | Superseeker | Maintained by N. J. A. Sloane (njas@research.att.com)

Last modified July 26 23:19 EDT 2008. Contains 142293 sequences.


AT&T Labs Research