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A131963
Expansion of f(x, x^2) * f(x^4, x^12) in powers of x where f(, ) is Ramanujan's general theta function.
11
1, 1, 1, 0, 1, 2, 1, 1, 0, 1, 0, 1, 2, 1, 1, 1, 1, 1, 0, 2, 0, 0, 1, 0, 2, 1, 3, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 2, 2, 1, 1, 0, 1, 1, 1, 2, 0, 0, 1, 1, 2, 0, 0, 2, 0, 1, 0, 1, 1, 2, 2, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 3, 0, 1, 0, 0, 1, 2, 2, 0, 1, 1, 2
OFFSET
0,6
COMMENTS
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
LINKS
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of psi(x^4) * phi(-x^3) / chi(-x) in powers of x where phi(), psi(), chi() are Ramanujan theta functions.
Expansion of q^(-13/24) * eta(q^2) * eta(q^3)^2 * eta(q^8)^2 / (eta(q) * eta(q^4) * eta(q^6)) in powers of q.
Euler transform of period 24 sequence [ 1, 0, -1, 1, 1, -1, 1, -1, -1, 0, 1, 0, 1, 0, -1, -1, 1, -1, 1, 1, -1, 0, 1, -2, ...].
a(25*n + 13) = a(n). a(25*n + 3) = a(25*n + 8) = a(25*n + 18) = a(25*n + 23) = 0.
2 * a(n) = A123484(24*n + 13).
EXAMPLE
G.f. = 1 + x + x^2 + x^4 + 2*x^5 + x^6 + x^7 + x^9 + x^11 + 2*x^12 + x^13 + ...
G.f. = q^13 + q^37 + q^61 + q^109 + 2*q^133 + q^157 + q^181 + q^229 + q^277 + ...
MATHEMATICA
a[ n_] := If[ n < 0, 0, With[ {m = 24 n + 13}, DivisorSum[ m, KroneckerSymbol[ -12, #] Mod[m/#, 2] &] / 2]]; (* Michael Somos, Nov 04 2015 *)
a[ n_] := SeriesCoefficient[(1/2) x^(-1/2) EllipticTheta[ 4, 0, x^3] QPochhammer[ -x, x] EllipticTheta[ 2, 0, x^2], {x, 0, n}]; (* Michael Somos, Nov 04 2015 *)
PROG
(PARI) {a(n) = if( n<0, 0, n = 24*n + 13; 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^3 + A)^2 * eta(x^8 + A)^2 / (eta(x + A) * eta(x^4 + A) * eta(x^6 + A)), n))};
CROSSREFS
Cf. A123484.
Sequence in context: A156709 A081400 A328194 * A130538 A276007 A351619
KEYWORD
nonn
AUTHOR
Michael Somos, Aug 02 2007
STATUS
approved