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A262146 Expansion of f(-x, -x^5) * f(x, x^7) / f(-x, -x^2)^2 in powers of x where f(, ) is Ramanujan's general theta function. 1
1, 2, 4, 8, 15, 25, 42, 68, 107, 166, 253, 377, 557, 811, 1166, 1661, 2344, 3275, 4543, 6253, 8544, 11600, 15653, 20994, 28011, 37178, 49100, 64550, 84489, 110115, 142951, 184867, 238196, 305844, 391391, 499244, 634865, 804925, 1017610, 1282957, 1613195 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
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 - (psi(x^6) / psi(x) - psi(x^6) / psi(-x)) / (2 * x) in powers of x^2 where psi() is a Ramanujan theta function.
Euler transform of period 48 sequence [ 2, 1, 2, 2, 1, 1, 2, 1, 3, 2, 1, 1, 1, 1, 3, 1, 2, 0, 1, 2, 2, 2, 2, 0, 2, 2, 2, 2, 1, 0, 2, 1, 3, 1, 1, 1, 1, 2, 3, 1, 2, 1, 1, 2, 2, 1, 2, 0, ...].
a(n) = A132217(2*n + 1) = - A262160(2*n + 1).
Convolution product of A097451 and A078408.
a(n) ~ exp(Pi*sqrt(n)) / (2^(7/2) * sqrt(3) * n^(3/4)). - Vaclav Kotesovec, Mar 31 2018
EXAMPLE
G.f. = 1 + 2*x + 4*x^2 + 8*x^3 + 15*x^4 + 25*x^5 + 42*x^6 + 68*x^7 + ...
G.f. = q^13 + 2*q^29 + 4*q^45 + 8*q^61 + 15*q^77 + 25*q^93 + 42*q^109 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ - x^(-5/8) EllipticTheta[ 2, 0, x^3] / EllipticTheta[ 2, 0, x^(1/2)], {x, 0, 2 n + 1}];
PROG
(PARI) {a(n) = my(A); if( n<0, 0, n = 2*n + 1; A = x * O(x^n); polcoeff( - eta(x + A) * eta(x^12 + A)^2 / (eta(x^2 + A)^2 * eta(x^6 + A)), n))};
CROSSREFS
Sequence in context: A325840 A347764 A324740 * A089140 A204555 A000125
KEYWORD
nonn
AUTHOR
Michael Somos, Oct 06 2015
STATUS
approved

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Last modified March 28 10:55 EDT 2024. Contains 371241 sequences. (Running on oeis4.)