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A128582
Expansion of f(x^4, x^12) * f(x, x^5) in powers of x where f(, ) is Ramanujan's general theta function.
11
1, 1, 0, 0, 1, 2, 0, 0, 1, 1, 0, 0, 2, 1, 0, 0, 1, 1, 0, 0, 2, 1, 0, 0, 1, 2, 0, 0, 1, 1, 0, 0, 1, 2, 0, 0, 0, 1, 0, 0, 3, 1, 0, 0, 1, 3, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 2, 1, 0, 0, 2, 2, 0, 0, 1, 2, 0, 0, 2, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 2, 1, 0, 0, 1, 2, 0
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) * psi(-x^3) / chi(-x^4)^2 in powers of x where psi(), chi() are Ramanujan theta functions.
Expansion of q^(-5/6) * eta(q^2)^2 * eta(q^3) * eta(q^8)^2 * eta(q^12) / (eta(q) * eta(q^4)^2 * eta(q^6)) in powers of q.
Euler transform of period 24 sequence [ 1, -1, 0, 1, 1, -1, 1, -1, 0, -1, 1, 0, 1, -1, 0, -1, 1, -1, 1, 1, 0, -1, 1, -2, ...].
G.f.: Product_{k>0} (1 + x^k) * (1 - x^(2*k)) * (1 + x^(4*k))^2 * (1 - x^(6*k - 3)) * (1 - x^(12*k)).
A128580(3*n + 2) = -2 * a(n). a(4*n) = A128583. a(4*n + 1) = A128591(n). a(4*n + 2) = a(4*n + 3) = 0.
EXAMPLE
G.f. = 1 + x + x^4 + 2*x^5 + x^8 + x^9 + 2*x^12 + x^13 + x^16 + x^17 + 2*x^20 + ...
G.f. = q^5 + q^11 + q^29 + 2*q^35 + q^53 + q^59 + 2*q^77 + q^83 + q^101 + q^107 + ...
MATHEMATICA
a[ n_] := If[ n < 0, 0, With[{m = 6 n + 5}, -1/2 DivisorSum[ m, KroneckerSymbol[ -12, #] KroneckerSymbol[ 2, m/#] &]]]; (* Michael Somos, Nov 15 2015 *)
a[ n_] := SeriesCoefficient[ 2^(-3/2) x^(-7/8) EllipticTheta[ 2, Pi/4, x^(3/2)] EllipticTheta[ 2, Pi/4, x]^2 / QPochhammer[ x], {x, 0, n}]; (* Michael Somos, Nov 15 2015 *)
PROG
(PARI) {a(n) = if( n<0, 0, n = 6*n + 5; -1/2 * sumdiv( n, d, kronecker( -12, d) * kronecker( 2, n/d)))};
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^3 + A) * eta(x^8 + A)^2 * eta(x^12 + A) / (eta(x + A) * eta(x^4 + A)^2 * eta(x^6 + A)), n))};
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael Somos, Mar 11 2007
STATUS
approved