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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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8
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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
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OFFSET
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0,3
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COMMENTS
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When multiplied by 16, this is the q-expansion of the automorphic function lambda (see A115977) [see Erdelyi].
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REFERENCES
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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
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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].
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FORMULA
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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
Expansion of eta(q)^8 * eta(q^4)^16 / eta(q^2)^24 in powers of q.
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. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = (1 / 16) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A128692. - Michael Somos, May 10 2014
G.f.: q * Product( (1 + q^(2*n)) / (1 + q^(2*n - 1)), n=1..inf )^8.
a(n) ~ (-1)^(n+1) * exp(2*Pi*sqrt(n))/(512*n^(3/4)). - Vaclav Kotesovec, Jul 10 2016
Empirical: Sum_{n>=0} a(n)/exp(2*Pi*n) = 17/16 - 3*sqrt(2)/4, verified to 27000 digits (10000 terms). - Simon Plouffe, Mar 01 2021
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EXAMPLE
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G.f. = 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
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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
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a[ n_] := SeriesCoefficient[ InverseEllipticNomeQ[ x] / 16, {x, 0, n}]; (* Michael Somos, Jun 13 2011 *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 2, 0, q] / EllipticTheta[ 2, 0, q^(1/2)])^8, {q, 0, n}]; (* Michael Somos, May 10 2014 *)
a[ n_] := SeriesCoefficient[ q (QPochhammer[ q^4] / QPochhammer[ -q])^8, {q, 0, n}]; (* Michael Somos, May 10 2014 *)
a[ n_] := SeriesCoefficient[ q (Product[ 1 - q^k, {k, 4, n - 1, 4}] / Product[ 1 - (-q)^k, {k, n - 1}])^8, {q, 0, n}]; (* Michael Somos, May 10 2014 *)
etr[p_] := Module[{b}, b[n_] := b[n] = If[n == 0, 1, Sum[Sum[d*p[d], {d, Divisors[ j]}]*b[n-j], {j, 1, n}]/n]; b]; aa = etr[Function[{n}, {-8, 16, -8, 0}[[Mod[n-1, 4]+1]]]]; a[n_] := aa[n-1]; Table[a[n], {n, 0, 26}] (* Jean-François Alcover, Mar 05 2015, after Alois P. Heinz *)
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PROG
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(PARI) {a(n) = my(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) = my(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
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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 a(x)=x*A001938(x)^2. Reversion of A005797.
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KEYWORD
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sign,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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