Search: a127393 -id:a127393
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A001938
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Expansion of k/(4*q^(1/2)) in powers of q, where k defined by sqrt(k) = theta_2(0, q)/theta_3(0, q).
(Formerly M3475 N1412)
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+10
22
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1, -4, 14, -40, 101, -236, 518, -1080, 2162, -4180, 7840, -14328, 25591, -44776, 76918, -129952, 216240, -354864, 574958, -920600, 1457946, -2285452, 3548550, -5460592, 8332425, -12614088, 18953310, -28276968, 41904208, -61702876, 90304598, -131399624
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OFFSET
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0,2
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COMMENTS
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k^2 is the parameter and q the Jacobi nome of elliptic functions. See, e.g., Fricke, p. 11, eq. (8) with p. 10. eq. (1). - Wolfdieter Lang, Jul 04 2016
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REFERENCES
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A. Cayley, A memoir on the transformation of elliptic functions, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 9, p. 128.
E. T. Copson, An Introduction to the Theory of Functions of a Complex Variable, 1935, Oxford University Press, p. 385.
N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; Eq. (34.3).
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
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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A. Cayley, A memoir on the transformation of elliptic functions, Philosophical Transactions of the Royal Society of London (1874): 397-456; Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, included in Vol. 9. [Annotated scan of pages 126-129]
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FORMULA
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Expansion of (psi(x^2) / phi(x))^2 = (psi(x) / phi(x))^4 = (psi(x^2) / psi(x))^4 = (psi(-x) / psi(-x^2))^4 = (chi(-x) / chi(-x^2)^2)^4 = (chi(x)^2 * chi(-x))^-4 = (chi(x) * chi(-x^2))^-4 = (f(-x^4) / f(x))^4 in powers of x where phi(), psi(), chi(), f() are Ramanujan theta functions. - Michael Somos, Feb 26 2012
G.f. A(x) satisfies 1 = (1 - 16 * x * A(x)^2) * (1 + 16 * x * A(-x)^2). - Michael Somos, Mar 26 2004
Expansion of q^(-1/2) * (eta(q) * eta(q^4)^2 / eta(q^2)^3)^4 in powers of q.
Euler transform of period 4 sequence [ -4, 8, -4, 0, ...].
Given g.f. A(x), then B(q) = q * A(q^2) satisfies 0 = f(B(q), B(q^2)) where f(u, v) = v - (u * (1 + 4*v))^2. - Michael Somos, Mar 26 2004
G.f. A(q) satisfies A(q) = sqrt(A(q^2)) / (1 + 4*q*A(q^2)); together with limit_{n->infinity} A(x^n) = 1 this gives a fast algorithm to compute the series. [Joerg Arndt, Aug 06 2011]
G.f.: (Product_{k>0} (1 + x^(2*k)) / (1 + x^(2*k - 1)))^4.
G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 1/4 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A139820. - Michael Somos, Jun 04 2015
G.f.: ((Sum_{n >= 0} x^(n*(n+1))) / (1 + Sum_{n >= 1} x^(n^2)))^4 (from the sum representation of the Jacobi theta functions evaluated at vanishing argument). - Wolfdieter Lang, Jul 04 2016
a(n) ~ (-1)^n * exp(sqrt(2*n)*Pi) / (32 * 2^(1/4) * n^(3/4)). - Vaclav Kotesovec, Nov 15 2017
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EXAMPLE
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G.f. = 1 - 4*x + 14*x^2 - 40*x^3 + 101*x^4 - 236*x^5 + 518*x^6 - 1080*x^7 + ...
G.f. of B(q) = q * A(q^2): q - 4*q^3 + 14*q^5 - 40*q^7 + 101*q^9 - 236*q^11 + 518*q^13 - 1080*q^15 + ...
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MATHEMATICA
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a[ n_] := SeriesCoefficient[ 1 / (QPochhammer[ -x, x^2] QPochhammer[ x^2, x^4])^4, {x, 0, n}]; (* Michael Somos, Sep 24 2011 *)
a[ n_] := SeriesCoefficient[ (QPochhammer[ x^4] / QPochhammer[ -x])^4, {x, 0, n}]; (* Michael Somos, Sep 24 2011 *)
a[ n_] := SeriesCoefficient[ (Product[ 1 - x^k, {k, 4, n, 4}] / Product[ 1 - (-x)^k, {k, n}])^4, {x, 0, n}]; (* Michael Somos, Sep 24 2011 *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 2, 0, q^(1/2)] / (2 EllipticTheta[ 3, 0, q]))^4, {q, 0, n + 1/2}]; (* Michael Somos, Sep 24 2011 *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 2, 0, q] / EllipticTheta[ 2, 0, q^(1/2)])^4, {q, 0, n + 1/2}]; (* Michael Somos, Sep 24 2011 *)
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PROG
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(PARI) {a(n) = my(A, A2, m); if( n<0, 0, n = 2*n + 1; A = x + O(x^3); m=2; while( m<n, m*=2; A = subst(A, x, x^2); A = sqrt(A) / (1 + 4*A)); polcoeff(A, n))}; /* Michael Somos, Mar 26 2004 */
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^4 + A)^2 / eta(x^2 + A)^3)^4, n))}; /* Michael Somos, Mar 26 2004 */
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CROSSREFS
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Cf. A000122, A000700, A001936, A010054, A079006, A093160, A121373, A127393, A127931, A127932, A139820.
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KEYWORD
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sign,easy,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A134746
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Expansion of 1+k in powers of q^(1/2) where q is Jacobi's nome and k is the elliptic modulus.
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+10
4
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1, 4, 0, -16, 0, 56, 0, -160, 0, 404, 0, -944, 0, 2072, 0, -4320, 0, 8648, 0, -16720, 0, 31360, 0, -57312, 0, 102364, 0, -179104, 0, 307672, 0, -519808, 0, 864960, 0, -1419456, 0, 2299832, 0, -3682400, 0, 5831784, 0, -9141808, 0, 14194200, 0, -21842368, 0
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OFFSET
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0,2
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COMMENTS
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LINKS
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FORMULA
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Expansion of (phi(q) / phi(q^2))^2 in powers of q where phi() is a Ramanujan theta function.
Expansion of (eta(q^8) / eta(q))^4 * (eta(q^2) / eta(q^4))^14 in powers of q.
Euler transform of period 8 sequence [ 4, -10, 4, 4, 4, -10, 4, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 2 g(t) where q = exp(2 Pi i t) and g() is the g.f. of A210066.
G.f.: ( (Sum_{k in Z} x^(k^2)) / (Sum_{k in Z} x^(2*k^2)) )^2 = ( Product_{k>0} (1 + x^k)^2 * (1 + x^(4*k))^2 / (1 + x^(2*k))^5 )^2.
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (2 - v)^2 - u * (2 - u) * v^2.
Empirical: Sum_{n>=0} a(n)/exp(2*Pi*n) = 34 + 24*sqrt(2) - 4*sqrt(140 + 99*sqrt(2)). - Simon Plouffe, Mar 04 2021
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EXAMPLE
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G.f. = 1 + 4*q - 16*q^3 + 56*q^5 - 160*q^7 + 404*q^9 - 944*q^11 + 2072*q^13 + ...
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MATHEMATICA
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CoefficientList[Series[(QPochhammer[x^8]/QPochhammer[x])^4 (QPochhammer[x^2]/QPochhammer[x^4])^14, {x, 0, 50}], x] (* Jan Mangaldan, Mar 21 2013 *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q] / EllipticTheta[ 3, 0, q^2])^2, {q, 0, n}]; (* Michael Somos, Oct 16 2015 *)
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PROG
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(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( ( (eta(x^8 + A) / eta(x + A))^2 * (eta(x^2 + A) / eta(x^4 + A))^7 )^2, n))};
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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A210067
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Expansion of (phi(-q) / phi(q^2))^2 in powers of q where phi() is a Ramanujan theta function.
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+10
3
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1, -4, 0, 16, 0, -56, 0, 160, 0, -404, 0, 944, 0, -2072, 0, 4320, 0, -8648, 0, 16720, 0, -31360, 0, 57312, 0, -102364, 0, 179104, 0, -307672, 0, 519808, 0, -864960, 0, 1419456, 0, -2299832, 0, 3682400, 0, -5831784, 0, 9141808, 0, -14194200, 0, 21842368, 0
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OFFSET
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0,2
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COMMENTS
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LINKS
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FORMULA
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Expansion of (eta(q)^2 * eta(q^2) * eta(q^8)^2 / eta(q^4)^5)^2 in powers of q.
Euler transform of period 8 sequence [ -4, -6, -4, 4, -4, -6, -4, 0, ...].
Empirical: Sum_{n>=0} a(n)/exp(2*Pi*n) = -32 - 24*sqrt(2) + 4*sqrt(140+99*sqrt(2)). - Simon Plouffe, Mar 02 2021
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EXAMPLE
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1 - 4*q + 16*q^3 - 56*q^5 + 160*q^7 - 404*q^9 + 944*q^11 - 2072*q^13 + ...
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MATHEMATICA
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a[n_] := SeriesCoefficient[(EllipticTheta[3, 0, -q]/EllipticTheta[3, 0, q^2])^2, {q, 0, n}]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Nov 29 2017 *)
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PROG
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(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A)^2 * eta(x^2 + A) * eta(x^8 + A)^2 / eta(x^4 + A)^5)^2, n))}
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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