A298116 Expansion of 1/q * chi(q) * chi(q^5) * chi(-q^20)^2 / chi(-q)^2 in powers of q where chi() is a Ramanujan theta function.

1, 3, 5, 10, 18, 30, 51, 80, 124, 190, 281, 410, 592, 840, 1178, 1640, 2253, 3070, 4154, 5570, 7422, 9830, 12932, 16920, 22028, 28520, 36761, 47180, 60280, 76720, 97278, 122880, 154693, 194110, 242776, 302740, 376424, 466710, 577114, 711800, 875707, 1074790

Offset

-1,2

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 = -1..1000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of 1/q * f(q) * f(q^5) / (phi(-q) * psi(q^10)) in powers of q where f(), phi(), psi() are Ramanujan theta functions.

Euler transform of period 20 sequence [3, -1, 3, 0, 4, -1, 3, 0, 3, -4, 3, 0, 3, -1, 4, 0, 3, -1, 3, 0, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (20 t)) = f(t) where q = exp(2 Pi i t).

a(n) = A058555(n) = A298107(n) unless n=0.

Expansion of (eta(q^2) * eta(q^10))^4/(eta(q^4)*eta(q^5)*(eta(q)* eta(q^20))^3) in powers of q. - G. C. Greubel, Mar 20 2018

a(n) ~ exp(2*Pi*sqrt(n/5)) / (2*5^(1/4)*n^(3/4)). - Vaclav Kotesovec, Mar 21 2018

Example

G.f. = q^-1 + 3 + 5*q + 10*q^2 + 18*q^3 + 30*q^4 + 51*q^5 + 80*q^6 + 124*q^7 + ...

Mathematica

a[ n_] := SeriesCoefficient[ 1/q QPochhammer[ q^10, q^20]^2 QPochhammer[-q, q]^2 QPochhammer[-q, q^2] QPochhammer[-q^5, q^10], {q, 0, n}];
eta[q_]:= q^(1/24)*QPochhammer[q]; a[n_]:= SeriesCoefficient[(eta[q^2]* eta[q^10])^4/(eta[q^4]*eta[q^5]*(eta[q]*eta[q^20])^3), {q, 0, n}]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Mar 20 2018 *)

Prog

(PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( eta(x^2 + A)^4 * eta(x^10 + A)^4 / (eta(x + A)^3 * eta(x^4 + A) * eta(x^5 + A) * eta(x^20 + A)^3), n))};

Crossrefs

Essentially the same as A058555 and A298107.

Sequence in context: A270414 A227208 A009854 * A357534 A018165 A054179

Adjacent sequences: A298113 A298114 A298115 * A298117 A298118 A298119

Keyword

nonn

Author

Michael Somos, Jan 12 2018

Status

approved