OFFSET
0,3
COMMENTS
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Eric Weisstein's World of Mathematics, Rogers Mod 14 Identities
FORMULA
Euler transform of period 14 sequence [ 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, ...]. - Michael Somos, Sep 21 2005
G.f.: Product_{k>0} (1 - x^(14*k)) * (1 - x^(14*k - 4)) * (1 - x^(14*k - 10)) / (1 - x^k) = Sum_{k>=0} x^(k^2 + k) / ((1 - x^(2*k + 1)) * Product_{j=1..k} (1 - x^j) * (1 - x^(2*j - 1))). - Michael Somos, Sep 21 2005
Expansion of f(-x^4, -x^10) / f(-x, -x^2) in powers of x where f(, ) is Ramanujan's general theta function. - Michael Somos, Nov 21 2015
Number of partitions of n into parts all not == 0, 4, 10 (mod 14). - Michael Somos, Nov 21 2015
a(n) ~ 11^(1/4) * cos(3*Pi/14) * exp(Pi*sqrt(11*n/21)) / (2 * 3^(1/4) * 7^(3/4) * n^(3/4)). - Vaclav Kotesovec, Nov 21 2015
EXAMPLE
G.f. = 1 + x + 2*x^2 + 3*x^3 + 4*x^4 + 6*x^5 + 9*x^6 + 12*x^7 + 17*x^8 + 23*x^29 + ...
G.f. = q^47 + q^215 + 2*q^383 + 3*q^551 + 4*q^719 + 6*q^887 + 9*q^1055 + 12*q^1223 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ QPochhammer[ x^4, x^14] QPochhammer[ x^10, x^14] QPochhammer[ x^14] / QPochhammer[ x], {x, 0, n}]; (* Michael Somos, Nov 21 2015 *)
a[ n_] := SeriesCoefficient[ Product[ (1 - x^k)^-{ 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0}[[Mod[k, 14, 1]]], {k, n}], {x, 0, n}]; (* Michael Somos, Nov 21 2015 *)
PROG
(PARI) {a(n) = if( n<0, 0, polcoeff( 1 / prod(k=1, n, 1 - [ 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1][k%14 + 1] * x^k, 1 + x * O(x^n)), n))}; /* Michael Somos, Sep 21 2005 */
CROSSREFS
KEYWORD
nonn
AUTHOR
Eric W. Weisstein, Apr 19 2005
STATUS
approved