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A005798
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Expansion of (theta_2(q)/theta_3(q))^4/16 in powers of q.
(Formerly M4528)
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6
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0, 1, -8, 44, -192, 718, -2400, 7352, -20992, 56549, -145008, 356388, -844032, 1934534, -4306368, 9337704, -19771392, 40965362, -83207976, 165944732, -325393024, 628092832, -1194744096, 2241688744, -4152367104, 7599231223, -13749863984
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Also, expansion of eta(q)^8 * eta(q^4)^16 / eta(q^2)^24 in powers of q.
When multiplied by 16, this is the q-expansion of the automorphic function lambda (see A119577) [see Erdelyi].
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
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REFERENCES
| M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math.Series 55, Tenth Printing, 1972, p. 591.
J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 121.
A. Erdelyi, Higher Transcendental Functions, McGraw-Hill, 1955, Vol. 3, p. 23, Eq. (37).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| Alois P. Heinz, Table of n, a(n) for n = 0..1000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math.Series 55, Tenth Printing, 1972, p. 591.
M. Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Eric Weisstein's World of Mathematics, Elliptic Lambda Function a section of The World of Mathematics.
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FORMULA
| Expansion of elliptic lambda / 16 = m / 16 = (k / 4)^2 in powers of the nome q.
Expansion of q * (psi(q) / phi(q))^8 = q * (psi(q^2) / psi(q))^8 = q * (psi(-q) / phi(-q^2))^8 = q * (chi(-q) / chi(-q^2)^2)^8 = q / (chi(q) * chi(-q^2))^8 = q / (chi(-q) * chi(q)^2)^8 = q * (psi(q^2) / phi(q))^4 = q * (f(-q^4) / f(q))^8 in powers of q where phi(), psi(), chi(), f() are Ramanujan theta functions. - Michael Somos Jun 13 2011
Euler transform of period 4 sequence [ -8, 16, -8, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u^2 - v + 16*u*v - 32*u^2*v + 256*(u*v)^2. - Michael Somos Mar 19 2004
G.f.: q * Product( (1 + q^(2*n)) / (1 + q^(2*n - 1)), n=1..inf )^8 = eta(q)^8 * eta(q^4)^16 / eta(q^2)^24; eta = Dedekind's function.
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EXAMPLE
| q - 8*q^2 + 44*q^3 - 192*q^4 + 718*q^5 - 2400*q^6 + 7352*q^7 - 20992*q^8 +
...
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MAPLE
| with (numtheory): etr:= proc(p) local b; b:=proc(n) option remember; local d, j; if n=0 then 1 else add (add (d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n fi end end: aa:=etr (n-> [ -8, 16, -8, 0] [modp(n-1, 4)+1]): a:= n->aa(n-1): seq (a(n), n=0..26); # Alois P. Heinz, Sep 08 2008
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MATHEMATICA
| a[ n_] := If[ n < 0, 0, SeriesCoefficient[ InverseEllipticNomeQ[ x] / 16, {x, 0, n}]] (* Michael Somos Jun 13 2011 *)
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PROG
| (PARI) {a(n) = local(A, m); if( n<1, 0, m=1; A = x + O(x^2); while( m<n, m*=2; A = sqrt( subst(A, x, x^2)); A /= (1 + 4*A)^2); polcoeff(A, n))}
(PARI) {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^4 + A)^2 / eta(x^2 + A)^3)^8, n))} /* Michael Somos Jul 16 2005 */
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CROSSREFS
| If initial 0 is omitted and sequence begins with a(0) = 1, then this is the convolution of A001938 with itself. G.f.s are related by A005798(x)=x*A001938(x)^2. Reversion of A005797.
Cf. A007248, A029845, A119577.
Sequence in context: A059596 A181358 * A092877 A160521 A023007 A169795
Adjacent sequences: A005795 A005796 A005797 * A005799 A005800 A005801
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KEYWORD
| sign,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| Simplified definition. - N. J. A. Sloane, Sep 25 2011
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