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Search: id:A002103
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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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3
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| 1, 2, 15, 150, 1707, 20910, 268616, 3567400, 48555069, 673458874, 9481557398, 135119529972, 1944997539623, 28235172753886, 412850231439153, 6074299605748746, 89857589279037102, 1335623521633805028
(list; graph; listen)
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OFFSET
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0,2
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REFERENCES
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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.
Z. X. Wang and D. R. Guo, Special Functions, World Scientific Publishing, 1989, page 512.
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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).
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EXAMPLE
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q = x + 2x^5 + 15x^9 + 150x^13 + ... where x = q - 2q^5 + 5q^9 - 10q^13 + ... coefficients from A079006.
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PROGRAM
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(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
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Cf. A001936, A002639.
Adjacent sequences: A002100 A002101 A002102 this_sequence A002104 A002105 A002106
Sequence in context: A111686 A001854 A060226 this_sequence A124548 A139085 A140809
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KEYWORD
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nonn,easy,nice
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AUTHOR
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njas
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