A281490 Expansion of f(x, x^3) * f(x, x^8) in powers of x where f(, ) is Ramanujan's general theta function.

1, 2, 1, 1, 1, 0, 1, 1, 1, 1, 1, 3, 1, 0, 2, 1, 1, 1, 1, 0, 0, 2, 1, 1, 0, 1, 2, 0, 2, 2, 1, 2, 1, 1, 0, 1, 3, 1, 0, 1, 2, 0, 0, 0, 1, 2, 2, 1, 0, 0, 0, 2, 1, 2, 1, 1, 2, 1, 2, 1, 0, 3, 0, 1, 1, 0, 4, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 1, 3, 1, 0, 0, 0, 0, 1, 3

Offset: 0

Author

Michael Somos, Jan 29 2017

Keywords

• nonn

Examples

			G.f. = 1 + 2*x + x^2 + x^3 + x^4 + x^6 + x^7 + x^8 + x^9 + x^10 + 3*x^11 + ...
G.f. = q^29 + 2*q^65 + q^101 + q^137 + q^173 + q^245 + q^281 + q^317 + q^353 + ...
		

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

Crossrefs

Cf. A010054, A281451, A281814.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ (1/2) x^(-1/8) EllipticTheta[ 2, 0, x^(1/2)] QPochhammer[ -x, x^9] QPochhammer[ -x^8, x^9] QPochhammer[ x^9], {x, 0, n}];

PARI

{a(n) = if( n<0, 0, sumdiv(36*n + 29, d, kronecker(-4, d)) / 2)};

PARI

{a(n) = if( n<0, 0, my(A, p, e); n = 36*n + 29; A = factor(n); prod(k=1, matsize(A) [1], [p, e] = A[k, ]; if(p%4==1, e+1, 1-e%2)) / 2)};

Formula

f(x,x^m) = 1 + Sum_{k=1..oo} x^((m+1)*k*(k-1)/2)*(x^k + x^(m*k)). - N. J. A. Sloane, Jan 30 2017

G.f.: (Sum_{k>0} x^(k*(k - 1)/2)) * (Sum_{k in Z} x^(k*(9*k + 7)/2)).

G.f.: Product_{k>0} (1 - x^(2*k)) / (1 - x^(2*k-1)) * (1 + x^(9*k-8)) * (1 + x^(9*k-1)) * (1 - x^(9*k)).

Convolution of sequences A010054 and A281814.

2 * a(n) = A281451(32*n + 25).