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A246863
Expansion of phi(x) * f(x^1, x^7) in powers of x where phi(), f() are Ramanujan theta functions.
2
1, 3, 2, 0, 2, 2, 0, 1, 2, 2, 3, 4, 0, 0, 2, 0, 4, 2, 0, 2, 0, 0, 1, 4, 0, 2, 6, 1, 2, 0, 0, 4, 2, 0, 0, 2, 4, 2, 2, 0, 0, 0, 0, 4, 0, 1, 4, 2, 0, 4, 2, 0, 3, 2, 2, 0, 4, 0, 2, 2, 0, 4, 0, 2, 2, 2, 0, 0, 2, 0, 2, 4, 0, 0, 2, 0, 3, 4, 0, 0, 2, 4, 2, 0, 0, 3, 4
OFFSET
0,2
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
Euler transform of period 16 sequence [ 3, -4, 2, -1, 2, -3, 3, -2, 3, -3, 2, -1, 2, -4, 3, -2, ...].
Convolution of A000122 and A214263.
a(9*n + 3) = a(9*n + 6) = 0. a(9*n) = A246862(n).
a(n) = A113407(2*n + 1) = - A226192(2*n + 1) = A008441(4*n + 2) = A134343(4*n + 2) = A116604(8*n + 4) = A125079(8*n + 4) = A129447(8*n + 4) = A138741(8*n + 4).
EXAMPLE
G.f. = 1 + 3*x + 2*x^2 + 2*x^4 + 2*x^5 + x^7 + 2*x^8 + 2*x^9 + 3*x^10 + ...
G.f. = q^9 + 3*q^25 + 2*q^41 + 2*q^73 + 2*q^89 + q^121 + 2*q^137 + 2*q^153 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x] QPochhammer[ -x^1, x^8] QPochhammer[ -x^7, x^8] QPochhammer[ x^8], {x, 0, n}];
PROG
(PARI) {a(n) = if( n<0, 0, issquare(16 * n + 9) + 2 * sum(i=1, sqrtint(n), issquare(16 * (n - i^2) + 9)))};
KEYWORD
nonn
AUTHOR
Michael Somos, Sep 05 2014
STATUS
approved