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A002103
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Coefficients of expansion of Jacobi nome q in powers of (1/2)(1-sqrt(k'))/(1+sqrt(k')).
(Formerly M2082 N0823)
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2
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1, 2, 15, 150, 1707, 20910, 268616, 3567400, 48555069, 673458874, 9481557398, 135119529972, 1944997539623, 28235172753886, 412850231439153, 6074299605748746, 89857589279037102, 1335623521633805028
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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REFERENCES
| Bramhall, J. N.; An iterative method for inversion of power series. Comm. ACM 4 1961 317-318.
H. R. P. Ferguson, D. E. Nielsen and G. Cook, A partition formula for the integer coefficients of the theta function nome, Math. Comp., 29 (1975), 851-855.
H. E. Fettis, Note on the computation of Jacobi's Nome and its inverse, Computing, 4 (1969), 202-206.
A. Fletcher, Guide to tables of elliptic functions, Math. Tables Other Aids Computation, 3 (1948), 229-281, Section III, p. 234. MR0030295 (10,741b)
A. N. Lowan, G. Blanch and W. Horenstein, On the inversion of the q-series associated with Jacobian elliptic functions, Bull. Amer. Math. Soc., 48 (1942), 737-738.
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).
Z. X. Wang and D. R. Guo, Special Functions, World Scientific Publishing, 1989, page 512.
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FORMULA
| 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).
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EXAMPLE
| q = x + 2x^5 + 15x^9 + 150x^13 + ... where x = q - 2q^5 + 5q^9 - 10q^13 + ... coefficients from A079006.
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MATHEMATICA
| 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}] (* From Jean-François Alcover, Nov 08 2011, after Michael Somos *)
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PROG
| (PARI) {a(n)=local(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))}
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CROSSREFS
| Cf. A001936, A002639.
Sequence in context: A111686 A001854 A060226 * A191364 A185756 A124548
Adjacent sequences: A002100 A002101 A002102 * A002104 A002105 A002106
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KEYWORD
| nonn,easy,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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