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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
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
-1,2
COMMENTS
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
LINKS
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
KEYWORD
nonn
AUTHOR
Michael Somos, Jan 12 2018
STATUS
approved

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Last modified May 19 13:25 EDT 2024. Contains 372694 sequences. (Running on oeis4.)