OFFSET
0,3
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^2) * phi(-x^3) / chi(-x) in powers of x where phi(), chi() are Ramanujan theta functions.
Expansion of q^(-1/24) * eta(q^3)^2 * eta(q^4)^5 / (eta(q) * eta(q^2) * eta(q^6) * eta(q^8)^2) in powers of q.
Euler transform of period 24 sequence [ 1, 2, -1, -3, 1, 1, 1, -1, -1, 2, 1, -4, 1, 2, -1, -1, 1, 1, 1, -3, -1, 2, 1, -2, ...].
a(25*n + 1) = a(n). a(25*n + 6) = a(25*n + 11) = a(25*n + 16) = a(25*n + 21) = 0.
a(n) = A123484(24*n + 1).
Expansion of phi(-x^3) * f(x^2)^2 / psi(-x) in powers of x where phi(), psi(), f() are Ramanujan theta functions. - Michael Somos, Nov 06 2015
EXAMPLE
G.f. = 1 + x + 3*x^2 + 2*x^3 + 2*x^4 + x^5 + 3*x^7 + 2*x^8 + 4*x^9 + 2*x^10 + ...
G.f. = q + q^25 + 3*q^49 + 2*q^73 + 2*q^97 + q^121 + 3*q^169 + 2*q^193 + 4*q^217 + ...
MATHEMATICA
a[ n_] := If[ n < 0, 0, With[ {m = 24 n + 1}, DivisorSum[ m, KroneckerSymbol[ -12, #] Mod[m/#, 2] &]]]; (* Michael Somos, Nov 06 2015 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x^3] EllipticTheta[ 3, 0, x^2] QPochhammer[ -x, x], {x, 0, n}]; (* Michael Somos, Nov 06 2015 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x^2] QPochhammer[ -x, x^3] QPochhammer[ -x^2, x^3] QPochhammer[ x^3], {x, 0, n}]; (* Michael Somos, Nov 06 2015 *)
PROG
(PARI) {a(n) = if( n<0, 0, n = 24*n + 1; sumdiv(n, d, kronecker( -12, d) * (n/d %2)))};
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^3 + A)^2 * eta(x^4 + A)^5 / (eta(x + A) * eta(x^2 + A) * eta(x^6 + A) * eta(x^8 + A)^2), n))};
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael Somos, Aug 02 2007
STATUS
approved