A281492 Expansion of f(x, x^3) * f(x^4, x^5) in powers of x where f(, ) is Ramanujan's general theta function.

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

Offset

0,6

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

Euler transform of period 18 sequence [1, -1, 1, 0, 2, -1, 1, -2, 0, -2, 1, -1, 2, 0, 1, -1, 1, -2, ...].

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

G.f.: Product_{k>0} (1 - x^(2*k)) / (1 - x^(2*k-1)) * (1 + x^(9*k-5)) * (1 + x^(9*k-4)) * (1 - x^(9*k)).

2 * a(n) = A281451(128*n + 17).

Example

G.f. = 1 + x + x^3 + x^4 + 2*x^5 + 2*x^6 + x^7 + x^8 + 2*x^10 + x^11 + ...
G.f. = q^5 + q^41 + q^113 + q^149 + 2*q^185 + 2*q^221 + q^257 + q^293 + ...

Mathematica

a[ n_] := SeriesCoefficient[ (1/2) x^(-1/8) EllipticTheta[ 2, 0, x^(1/2)] QPochhammer[ -x^4, x^9] QPochhammer[ -x^5, x^9] QPochhammer[ x^9], {x, 0, n}];

Prog

(PARI) {a(n) = if( n<0, 0, sumdiv(36*n + 5, d, kronecker(-4, d)) / 2)};

Crossrefs

Cf. A281451.

Sequence in context: A185304 A081389 A133685 * A112183 A275451 A269317

Adjacent sequences: A281489 A281490 A281491 * A281493 A281494 A281495

Keyword

nonn

Author

Michael Somos, Jan 29 2017

Status

approved