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A262984
Expansion of f(-x^2, -x^10) / f(-x, -x) in powers of x where f(, ) is Ramanujan's general theta function.
1
1, 2, 3, 6, 10, 16, 26, 40, 60, 90, 131, 188, 268, 376, 522, 720, 983, 1330, 1790, 2390, 3170, 4184, 5488, 7160, 9300, 12020, 15466, 19822, 25300, 32168, 40760, 51464, 64763, 81250, 101620, 126726, 157604, 195472, 241810, 298400, 367340, 451156, 552867, 676030
OFFSET
0,2
COMMENTS
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
REFERENCES
Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 6, 7th equation.
LINKS
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of psi(x^6) * phi(-x^2) / f(-x)^2 in powers of x where phi(), psi(), f() are Ramanujan theta functions.
Expansion of q^(-2/3) * eta(q^2)^2 * eta(q^12)^2 / (eta(q)^2 * eta(q^4) * eta(q^6)) in powers of q.
Euler transform of period 12 sequence [ 2, 0, 2, 1, 2, 1, 2, 1, 2, 0, 2, 0, ...].
-2 * a(n) = A262967(3*n + 2).
a(n) ~ 5^(1/4) * exp(sqrt(5*n/6)*Pi) / (2^(13/4) * 3^(3/4) * n^(3/4)). - Vaclav Kotesovec, Oct 06 2015
G.f.: Sum_{k>=0} x^k * (Product_{i=1..k} 1 + x^(2*i)) / Product_{i=1..2*k+1} 1 - x^i). [Ramanujan] - Michael Somos, Nov 18 2015
EXAMPLE
G.f. = 1 + 2*x + 3*x^2 + 6*x^3 + 10*x^4 + 16*x^5 + 26*x^6 + 40*x^7 + ...
G.f. = q^2 + 2*q^5 + 3*q^8 + 6*q^11 + 10*q^14 + 16*q^17 + 26*q^20 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, x^3] EllipticTheta[ 4, 0, x^2] / (2 x^(3/4) QPochhammer[ x]^2), {x, 0, n}];
nmax=60; CoefficientList[Series[Product[(1-x^(12*k)) * (1+x^(6*k)) * (1+x^(2*k-1)) / (1-x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 06 2015 *)
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^12 + A)^2 / (eta(x + A)^2 * eta(x^4 + A) * eta(x^6 + A)), n))};
(PARI) q='q+O('q^99); Vec(eta(q^2)^2*eta(q^12)^2/(eta(q)^2*eta(q^4)*eta(q^6))) \\ Altug Alkan, Mar 19 2018
CROSSREFS
Cf. A262967.
Sequence in context: A280908 A146163 A101277 * A201077 A355383 A023655
KEYWORD
nonn
AUTHOR
Michael Somos, Oct 06 2015
STATUS
approved