OFFSET
1,2
COMMENTS
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..2500
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of q * phi(q) * psi(q^2)^4 in powers of q where phi(), psi() are Ramanujan theta functions.
Expansion of eta(q^2) * eta(q^4)^6 / eta(q)^2 in powers of q.
Euler transform of period 4 sequence [2, 1, 2, -5, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = 2^(-3/2) (t/I)^(5/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A245643.
G.f.: x * Product_{k>0} (1 + x^k) * (1 - x^(4*k))^6 / (1 - x^k).
Convolution inverse of A134414.
EXAMPLE
G.f. = x + 2*x^2 + 4*x^3 + 8*x^4 + 8*x^5 + 12*x^6 + 16*x^7 + 16*x^8 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ q QPochhammer[ q^4]^6 / EllipticTheta[ 4, 0, q], {q, 0, n}];
a[ n_] := SeriesCoefficient[ 2^-4 EllipticTheta[ 3, 0, q] EllipticTheta[ 2, 0, q]^4, {q, 0, n}];
PROG
(PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^4 + A)^6 / eta(x + A)^2, n))};
(Magma) A := Basis( ModularForms( Gamma1(4), 5/2), 59); A[2];
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael Somos, Jan 03 2016
STATUS
approved