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A002103 Coefficients of expansion of Jacobi nome q in certain powers of (1/2)*(1 - sqrt(k')) / (1 + sqrt(k')).
(Formerly M2082 N0823)
13
1, 2, 15, 150, 1707, 20910, 268616, 3567400, 48555069, 673458874, 9481557398, 135119529972, 1944997539623, 28235172753886, 412850231439153, 6074299605748746, 89857589279037102, 1335623521633805028 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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/n) tends to 16. - Vaclav Kotesovec, Jul 02 2016

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

REFERENCES

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.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..200

J. N. Bramhall, 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)

R. Fricke, Die elliptischen Funktionen und ihre Anwendungen, Dritter Teil, Springer-Verlag, 2012, p. 3, eq. (6)

C. Hermite, Annotated scan of a page from the Oeuvres, together with a page from Math. Tables Aids Comp., Vol. 3, 1948 that refers to it.

A. Kneser, Neue Untersuchung einer Reihe aus der Theorie der elliptischen Funktionen, J. reine u. angew. Math. 157, 1927, 209 - 218, p.218.

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.

H. P. Robinson, Letter to N. J. A. Sloane, Oct 07, 1976

R. E. Shafer, Review of Fettis (1969), Computing Reviews, July 1970, page 401 [Annotated scanned copy]

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).

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

EXAMPLE

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 + ...).

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}] (* Jean-Fran├žois 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 *)

PROG

(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

CROSSREFS

Cf. A001936, A002639, A079006.

Sequence in context: A111686 A001854 A060226 * A191364 A233832 A185756

Adjacent sequences:  A002100 A002101 A002102 * A002104 A002105 A002106

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified October 23 01:36 EDT 2018. Contains 316518 sequences. (Running on oeis4.)