A240948 Expansion of chi(x^5)^6 + x * chi(x)^6 in powers of x where chi() is a Ramanujan theta function.

1, 1, 6, 15, 26, 57, 102, 172, 276, 453, 743, 1128, 1698, 2539, 3780, 5531, 7882, 11238, 15918, 22259, 30861, 42438, 58110, 78909, 106392, 142872, 190698, 253179, 334266, 439581, 575956, 750613, 974316, 1260336, 1624702, 2086806, 2670162, 3406695, 4333590

Offset

0,3

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

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

Expansion of (chi(x) * chi(x^5))^6 - 4 * x * chi(x) * chi(x^5) in powers of x where chi() is a Ramanujan theta function.

Expansion of (eta(q^2)^2 / (eta(q) * eta(q^4)))^6 + (eta(q^10)^2 / (eta(q^5) * eta(q^20)))^6 in powers of q.

G.f. is a period 1 Fourier series which satisfies f(-1 / (320 t)) = f(t) where q = exp(2 Pi i t).

a(n) ~ exp(Pi*sqrt(n)) / (2^(3/2) * n^(3/4)). - Vaclav Kotesovec, Nov 29 2019

Example

G.f. = 1 + x + 6*x^2 + 15*x^3 + 26*x^4 + 57*x^5 + 102*x^6 + 172*x^7 + ...
G.f. = q^-5 + q^-1 + 6*q^3 + 15*q^7 + 26*q^11 + 57*q^15 + 102*q^19 + ...

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ x^5, -x^5]^-6 + x QPochhammer[ x, -x]^-6, {x, 0, n}];
nmax = 40; CoefficientList[Series[x*Product[((1 + x^k)/(1 + x^(2*k)))^6, {k, 1, nmax}] + Product[((1 + x^(5*k))/(1 + x^(10*k)))^6, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 29 2019 *)

Prog

(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( x * (eta(x^2 + A)^2 / (eta(x + A) * eta(x^4 + A)))^6 + (eta(x^10 + A)^2 / (eta(x^5 + A) * eta(x^20 + A)))^6, n))};
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); A = eta(x^2 + A)^2 * eta(x^10 + A)^2 / (eta(x + A) * eta(x^4 + A) * eta(x^5 + A) * eta(x^20 + A)); polcoeff( A^5 - 4 * x * A, n))};

Crossrefs

Sequence in context: A284629 A022601 A112150 * A072257 A227952 A373384

Adjacent sequences: A240945 A240946 A240947 * A240949 A240950 A240951

Keyword

nonn

Author

Michael Somos, Aug 04 2014

Status

approved