OFFSET
0,6
COMMENTS
REFERENCES
Evans (1990) page 99 equation (1.10) is the connection between A214129, A214130, A214131. - Michael Somos, Nov 01 2024
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Ronald J. Evans, Theta Function Identities, J. of Mathematical Analysis and Applications 147(1990), 97-121.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of f(-x^13)^2 / (f(-x, -x^12) * f(-x^5, -x^8)) in powers of x where f() is Ramanujan's two-variable theta function.
Euler transform of period 13 sequence [ 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, ...].
G.f.: 1 / (Product_{k>0} (1 - x^(13*k - 1)) * (1 - x^(13*k - 5)) * (1 - x^(13*k - 8)) * (1 - x^(13*k - 12))).
EXAMPLE
G.f. = 1 + x + x^2 + x^3 + x^4 + 2*x^5 + 2*x^6 + 2*x^7 + 3*x^8 + 3*x^9 + 4*x^10 + ...
G.f. = q^-1 + q^5 + q^11 + q^17 + q^23 + 2*q^29 + 2*q^35 + 2*q^41 + 3*q^47 + 3*q^53 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ 1 / (QPochhammer[ q, q^13] QPochhammer[ q^5, q^13] QPochhammer[ q^8, q^13] QPochhammer[ q^12, q^13]), {q, 0, n}]
PROG
(PARI) {a(n) = if( n<0, 0, polcoeff( 1 / prod( k=1, n, 1 - [ 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1][k%13 + 1] * x^k, 1 + x * O(x^n)), n))}
CROSSREFS
KEYWORD
nonn,changed
AUTHOR
Michael Somos, Jul 04 2012
STATUS
approved