A281640 Expansion of x * f(x, x) * f(x^5, x^25) in powers of x where f(, ) is Ramanujan's general theta function.
1, 2, 0, 0, 2, 1, 2, 0, 0, 4, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 3, 2, 0, 0, 2, 2, 0, 0, 0, 2, 0, 2, 0, 0, 0, 1, 6, 0, 0, 2, 0, 0, 0, 0, 4, 2, 0, 0, 0, 2, 0, 2, 0, 0, 0, 0, 2, 0, 0, 2, 2, 0, 0, 0, 2, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 1, 4, 0, 0, 2, 0, 2
Offset: 1
Keywords
Examples
G.f. = x + 2*x^2 + 2*x^5 + x^6 + 2*x^7 + 4*x^10 + 2*x^15 + 2*x^17 + 2*x^22 + ... G.f. = q^5 + 2*q^8 + 2*q^17 + q^20 + 2*q^23 + 4*q^32 + 2*q^47 + 2*q^53 + 2*q^65 + ...
Links
- G. C. Greubel, Table of n, a(n) for n = 1..1000
- Michael Somos, Introduction to Ramanujan theta functions
- Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Programs
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Mathematica
a[ n_] := If[ n < 1, 0, DivisorSum[ 3 n + 2, KroneckerSymbol[ -15, #] (-1)^Boole[Mod[#, 4] == 2] &]]; a[ n_] := SeriesCoefficient[ x EllipticTheta[ 3, 0, x] QPochhammer[ -x^5, x^30] QPochhammer[ -x^25, x^30] QPochhammer[ x^30], {x, 0, n}]; -
PARI
{a(n) = if( n<1, 0, sumdiv(3*n + 2, d, kronecker(-15, d) * (-1)^(d%4==2) ))}; -
PARI
{a(n) = if( n<1, 0, my(m = 3*n + 2, s, x); for(y=1, sqrtint(m\5), if( y%3 && issquare((m - 5*y^2)\3, &x), s += (x>0) + 1)); s)}; -
PARI
{a(n) = if( n<1, 0, my(A); n--; A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^10 + A)^2 * eta(x^15 + A) * eta(x^60 + A) / (eta(x + A)^2 * eta(x^4 + A)^2 * eta(x^5 + A) * eta(x^20 + A) * eta(x^30 + A)), n))};
Formula
G.f.: x * (Sum_{k in Z} x^k^2) * (Sum_{k in Z} x^(15*k^2 - 10*k)).
G.f.: x * Product_{k>0} (1 + x^(2*k-1))^2 * (1 - x^(2*k)) * (1 + x^(30*k-25)) * (1 + x^(30*k-5)) * (1 - x^(30*k)).