%I A002103 M2082 N0823
%S A002103 1,2,15,150,1707,20910,268616,3567400,48555069,673458874,9481557398,
%T A002103 135119529972,1944997539623,28235172753886,412850231439153,
%U A002103 6074299605748746,89857589279037102,1335623521633805028
%N A002103 Coefficients of expansion of Jacobi nome q in powers of (1/2)(1-sqrt(k'))/
               (1+sqrt(k')).
%D A002103 Bramhall, J. N.; An iterative method for inversion of power series. Comm. 
               ACM 4 1961 317-318.
%D A002103 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.
%D A002103 H. E. Fettis, Note on the computation of Jacobi's Nome and its inverse, 
               Computing, 4 (1969), 202-206.
%D A002103 A. Fletcher, Guide to tables of elliptic functions, Math. Tables Other 
               Aids Computation, 3 (1948), 229-281, Section III, p. 234. MR0030295 
               (10,741b)
%D A002103 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.
%D A002103 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A002103 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A002103 Z. X. Wang and D. R. Guo, Special Functions, World Scientific Publishing, 
               1989, page 512.
%F A002103 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).
%e A002103 q = x + 2x^5 + 15x^9 + 150x^13 + ... where x = q - 2q^5 + 5q^9 - 10q^13 
               + ... coefficients from A079006.
%o A002103 (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))}
%Y A002103 Cf. A001936, A002639.
%Y A002103 Sequence in context: A111686 A001854 A060226 this_sequence A124548 A139085 
               A140809
%Y A002103 Adjacent sequences: A002100 A002101 A002102 this_sequence A002104 A002105 
               A002106
%K A002103 nonn,easy,nice
%O A002103 0,2
%A A002103 N. J. A. Sloane (njas(AT)research.att.com).