OFFSET
0,4
COMMENTS
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of phi(-x^4) * psi(-x^6) / chi(-x) in powers of x where phi(), chi() are Ramanujan theta functions.
Expansion of q^(-19/24) * eta(q^2) * eta(q^4)^2 * eta(q^6) * eta(q^24) / (eta(q) * eta(q^8) * eta(q^12)) in powers of q.
Euler transform of period 24 sequence [ 1, 0, 1, -2, 1, -1, 1, -1, 1, 0, 1, -2, 1, 0, 1, -1, 1, -1, 1, -2, 1, 0, 1, -2, ...].
a(25*n + 19) = a(n). a(25*n + 4) = a(25*n + 9) = a(25*n + 14) = a(25*n + 24) = 0.
2 * a(n) = A123484(24*n + 19).
EXAMPLE
G.f. = 1 + x + x^2 + 2*x^3 + x^5 + x^6 + x^8 + 2*x^10 + x^11 + x^12 + x^13 + ...
G.f. = q^19 + q^43 + q^67 + 2*q^91 + q^139 + q^163 + q^211 + 2*q^259 + q^283 + ...
MATHEMATICA
a[ n_] := If[ n < 0, 0, With[ {m = 24 n + 19}, DivisorSum[ m, KroneckerSymbol[ -12, #] Mod[m/#, 2] &] / 2]]; (* Michael Somos, Nov 03 2015 *)
a[ n_] := SeriesCoefficient[ 2^(-1/2) x^(-3/4) EllipticTheta[ 4, 0, x^4] QPochhammer[ -x, x] EllipticTheta[ 2, Pi/4, x^3], {x, 0, n}]; (* Michael Somos, Nov 03 2015 *)
PROG
(PARI) {a(n) = if( n<0, 0, n = 24*n + 19; sumdiv(n, d, kronecker( -12, d) * (n/d %2)) / 2)};
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^4 + A)^2 * eta(x^6 + A) * eta(x^24 + A) / (eta(x + A) * eta(x^8 + A) * eta(x^12 + A)), n))};
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael Somos, Aug 02 2007
STATUS
approved