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A002103
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Coefficients of expansion of Jacobi nome q in certain powers of (1/2)*(1 - sqrt(k')) / (1 + sqrt(k')).
(Formerly M2082 N0823)
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13
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1, 2, 15, 150, 1707, 20910, 268616, 3567400, 48555069, 673458874, 9481557398, 135119529972, 1944997539623, 28235172753886, 412850231439153, 6074299605748746, 89857589279037102, 1335623521633805028
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OFFSET
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0,2
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COMMENTS
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The Fricke reference has equation q^(1/4) = (sqrt(k) / 2) + 2(sqrt(k) / 2)^5 + 15(sqrt(k) / 2)^9 + 150(sqrt(k) / 2)^13 + 1707(sqrt(k) / 2)^17 + ... - Michael Somos, Jul 13 2013
a(n-1) appears in the expansion of the Jacobi nome q as q = x*Sum_{n >= 1} a(n-1)*x^(4*n) with x = (1/2)*(1 - sqrt(k')) / (1 + sqrt(k')), with the complementary modulus k' of elliptic functions. See, e.g., the Fricke, Kneser and Tricomi references, and the g.f. with example below. - Wolfdieter Lang, Jul 09 2016
The King-Canfield (1992) reference shows how this sequence is used in real life - it is one of the ingredients in solving the general quintic equation using elliptic functions. - N. J. A. Sloane, Dec 24 2019
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REFERENCES
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King, R. B., and E. R. Canfield. "Icosahedral symmetry and the quintic equation." Computers & Mathematics with Applications 24.3 (1992): 13-28. See Eq. (4.28).
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).
F. Tricomi, Elliptische Funktionen (German translation by M. Krafft of: Funzioni ellittiche), Akademische Verlagsgesellschaft Geest & Portig K.-G., Leipzig, 1948, p. 176, eq. (3.88).
Z. X. Wang and D. R. Guo, Special Functions, World Scientific Publishing, 1989, page 512.
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LINKS
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FORMULA
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a(n) = Sum {1<=k<=n} (-1)^k Sum { (4n+k)! C_1^b_1 ... C_n^b_n / (4n+1)! b_1! ... b_n! }, where the inner sum is over all partitions k = b_1 + ... + b_n, n = Sum i*b_i, b_i >= 0 and C_0=1, C_1=-2, C_2=5, C_3=-10 ... is given by (-1)^n*A001936(n).
G.f.: Series_Reversion( (theta_3(x) - theta_3(-x)) / (4*theta_3(x^4)) ) = Sum_{n>=0} a(n)*x^(4*n+1), where theta_3(x) = 1 + 2*Sum_{n>=1} x^(n^2). - Paul D. Hanna, Jan 07 2014
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EXAMPLE
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G.f. = 1 + 2*x + 15*x^2 + 150*x^3 + 1707*x^4 + 20910*x^5 + 268616*x^6 + 3567400*x^7 + ...
Jacobi nome q = x + 2x^5 + 15x^9 + 150x^13 + ... where x = q - 2q^5 + 5q^9 - 10q^13 + ... coefficients from A079006.
The series reversion of q = x + 2*x^5 + 15*x^9 + 150*x^13 + 1707*x^17 + ... equals (x + x^9 + x^25 + x^49 + ...)/(1 + 2*x^4 + 2*x^16 + 2*x^36 + 2*x^64 + ...).
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MATHEMATICA
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max = 18; A079006[n_] := SeriesCoefficient[ Product[(1+x^(k+1)) / (1+x^k), {k, 1, n, 2}]^2, {x, 0, n}]; A079006[0] = 1; sq = Series[ Sum[ A079006[n]*q^(4n+1), {n, 0, max}], {q, 0, 4max}]; coes = CoefficientList[ InverseSeries[ sq, x], x]; a[n_] := coes[[4n + 2]]; Table[a[n], {n, 0, max-1}] (* Jean-François Alcover, Nov 08 2011, after Michael Somos *)
a[ n_] := If[ n < 0, 0, SeriesCoefficient[ (EllipticNomeQ[ 16 x] / x)^(1/4), {x, 0, n}]]; (* Michael Somos, Jul 13 2013 *)
a[ n_] := With[{m = 4 n + 1}, If[ n < 0, 0, SeriesCoefficient[ InverseSeries[ Series[ q (QPochhammer[ q^16] / QPochhammer[-q^4])^2, {q, 0, m}], x], {x, 0, m}]]]; (* Michael Somos, Jul 13 2013 *)
a[ n_] := With[{m = 4 n + 1}, SeriesCoefficient[ InverseSeries[ Series[ 1/2 EllipticTheta[ 2, 0, x^4] / EllipticTheta[ 3, 0, x^4], {x, 0, m}]], {x, 0, m}]]; (* Michael Somos, Apr 14 2015 *)
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PROG
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(PARI) {a(n) = my(A); if( n<0, 0, n = 4*n + 1; A = O(x^n); polcoeff( serreverse( x * (eta(x^4 + A) * eta(x^16 + A)^2 / eta(x^8 + A)^3)^2), n))};
(PARI) {a(n)=local(A, N=sqrtint(n)+1); A=serreverse(sum(n=1, N, x^((2*n-1)^2))/(1+2*sum(n=1, N, x^(4*n^2)) +O(x^(4*n+4)))); polcoeff(A, 4*n+1)} \\ Paul D. Hanna, Jan 07 2014
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CROSSREFS
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KEYWORD
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nonn,easy,nice
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AUTHOR
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STATUS
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approved
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