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

1,2

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

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)).

a(n) = A122855(3*n + 2) = A260649(3*n + 2) = A122856(6*n + 4) = A258276(6*n + 4).

a(n) = - A140727(3*n + 2). 2 * a(n) = A192323(3*n + 2).

Example

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 + ...

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}];

Prog

(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))};

Crossrefs

Cf. A122855, A122856, A140727, A192323, A258276, A260649.

Sequence in context: A029303 A361163 A361500 * A281452 A341023 A241067

Adjacent sequences: A281637 A281638 A281639 * A281641 A281642 A281643

Keyword

nonn

Author

Michael Somos, Jan 25 2017

Status

approved