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A259895
Expansion of psi(x^2) * psi(x^3) in powers of x where psi() is a Ramanujan theta function.
6
1, 0, 1, 1, 0, 1, 1, 0, 0, 2, 0, 1, 1, 0, 0, 2, 0, 0, 1, 0, 2, 1, 0, 1, 1, 0, 0, 0, 0, 1, 3, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 2, 0, 0, 2, 0, 1, 1, 0, 1, 2, 0, 0, 0, 0, 1, 1, 0, 1, 2, 0, 0, 1, 0, 3, 0, 0, 0, 1, 0, 0, 2, 0, 1, 3, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 2
OFFSET
0,10
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 q^(-5/8) * eta(q^4)^2 * eta(q^6)^2 / (eta(q^2) * eta(q^3)) in powers of q.
Euler transform of period 12 sequence [ 0, 1, 1, -1, 0, 0, 0, -1, 1, 1, 0, -2, ...].
a(n) = A259896(3*n + 1). a(3*n) = A128583(n). a(3*n + 1) = a(9*n + 8) = 0.
2 * a(n) = A129402(4*n + 2) = A190615(4*n + 2) = A000377(8*n + 5) = A192013(8*n + 5). - Michael Somos, Jul 22 2015
-2 * a(n) = A259668(2*n + 1) = A128580(4*n + 2) = A134177(4*n + 2) = A257921(6*n + 3). - Michael Somos, Jul 22 2015
a(3*n + 2) = A259896(n). - Michael Somos, Jul 22 2015
EXAMPLE
G.f. = 1 + x^2 + x^3 + x^5 + x^6 + 2*x^9 + x^11 + x^12 + 2*x^15 + x^18 + ...
G.f. = q^5 + q^21 + q^29 + q^45 + q^53 + 2*q^77 + q^93 + q^101 + 2*q^125 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, x] EllipticTheta[ 2, 0, x^(3/2)] / (4 q^(5/8)), {x, 0, n}];
a[ n_] := If[ n < 0, 0, 1/2 Sum[ KroneckerSymbol[ -6, d], {d, Divisors[8 n + 5]}]]; (* Michael Somos, Jul 22 2015 *)
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^4 + A)^2 * eta(x^6 + A)^2 / (eta(x^2 + A) * eta(x^3 + A)), n))};
(PARI) {a(n) = if( n<0, 0, 1/2 * sumdiv( 8*n + 5, d, kronecker( -6, d)))};
KEYWORD
nonn
AUTHOR
Michael Somos, Jul 07 2015
STATUS
approved