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

0,2

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

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).

Example

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 + ...

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}];

Prog

(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)};

Crossrefs

Cf. A010054, A281451, A281814.

Sequence in context: A367622 A368647 A240009 * A326695 A343749 A323191

Adjacent sequences: A281487 A281488 A281489 * A281491 A281492 A281493

Keyword

nonn

Author

Michael Somos, Jan 29 2017

Status

approved