OFFSET
0,2
COMMENTS
REFERENCES
A. Cayley, An Elementary Treatise on Elliptic Functions, 2nd ed, 1895, p. 380, Section 488.
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
J. H. Conway and S. P. Norton, Monstrous Moonshine, Bull. Lond. Math. Soc. 11 (1979) 308-339. See page 336.
D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).
J. McKay and A. Sebbar, Fuchsian groups, automorphic functions and Schwarzians, Math. Ann., 318 (2000), 255-275.
J. McKay and H. Strauss, The q-series of monstrous moonshine and the decomposition of the head characters, Comm. Algebra 18 (1990), no. 1, 253-278.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
G.f.: ( Product_{k>0} (1 + q^(2*k - 1)) / (1 + q^(2*k)) )^4.
Expansion of q^(1/4) * (1 + k) / k^(1/2) in powers of q^(1/2) where q is Jacobi's nome and k is the elliptic modulus. - Michael Somos, Aug 01 2011
Expansion of q^(1/2) * 4 / k in powers of q where q is Jacobi's nome and k is the elliptic modulus. - Michael Somos, Aug 01 2011 and Feb 28 2012
Expansion of (phi(x) / psi(x))^4 in powers of x where phi(), psi() are Ramanujan theta functions.
Expansion of q^(1/2) * (eta(q^2)^3 / (eta(q) * eta(q^4)^2))^4 in powers of q. - Michael Somos, Aug 01 2011
Euler transform of period 4 sequence [4, -8, 4, 0, ...]. - Michael Somos, Mar 18 2004
Given g.f. A(x), then B(q) = A(q^2) / q satisfies 0 = f(B(q), B(q^2)) where f(u, v) = 16 + 8*v + v^2 - u^2*v. - Michael Somos, Mar 18 2004
G.f. A(q) satisfies A(q) = sqrt(A(q^2))+4*q/sqrt(A(q^2)). - Joerg Arndt, Aug 06 2011
A112143(n) = (-1)^n * a(n). a(2*n) = A029839(n). a(2*n + 1) = 4 * A079006(n). - Michael Somos, Mar 27 2004.
EXAMPLE
G.f. = 1 + 4*x + 2*x^2 - 8*x^3 - x^4 + 20*x^5 - 2*x^6 - 40*x^7 + 3*x^8 + ...
T8E = 1/q + 4*q + 2*q^3 - 8*q^5 - q^7 + 20*q^9 - 2*q^11 - 40*q^13 + 3*x^15 + ...
MATHEMATICA
a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ 4 / Sqrt[m], {q, 0, n - 1/2}]]; (* Michael Somos, Aug 01 2011 *)
a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ 2 (1 + Sqrt[m]) / m^(1/4), {q, 0, n/2 - 1/4}]]; (* Michael Somos, Aug 01 2011 *)
a[ n_] := SeriesCoefficient[ (QPochhammer[ x^2]^3 / (QPochhammer[ x] QPochhammer[x^4]^2))^4, {x, 0, n}]; (* Michael Somos, Aug 20 2014 *)
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^3 / (eta(x + A) * eta(x^4 + A))^2)^4, n))};
(PARI) {a(n) = my(A, m); if( n<0, 0, A = 1 + O(x); m=1; while( m<=n, m*=2; A = subst(A, x, x^2); A = (4*x + A) / sqrt(A)); polcoeff(A, n))};
CROSSREFS
KEYWORD
sign,easy,nice
AUTHOR
STATUS
approved