OFFSET
0,8
COMMENTS
Case k=3,i=1 of Gordon/Goellnitz/Andrews Theorem.
Also number of partitions in which no odd part is repeated, with no part of size less than or equal to 2 and where differences between parts at distance 2 are greater than 1 when the larger part is odd and greater than 2 when the larger part is even.
Euler transform of period 12 sequence [0,0,1,1,1,0,1,1,1,0,0,0,...]. - Michael Somos, Jun 28 2004
REFERENCES
G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 114.
FORMULA
a(n) ~ exp(Pi*sqrt(n/3)) * sqrt(2*sqrt(3) - 3) / (12 * n^(3/4)). - Vaclav Kotesovec, May 10 2018
MATHEMATICA
nmax = 60; CoefficientList[Series[Product[(1 - x^(4*k - 2))*(1 - x^(12*k))*(1 - x^(12*k+1-12))*(1 - x^(12*k- 1))/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, May 10 2018 *)
PROG
(PARI) {a(n)=if(n<0, 0, polcoeff(1/prod(k=1, n, 1-([0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0, 0][(k-1)%12+1])*x^k, 1+x*O(x^n)), n))} /* Michael Somos, Jun 28 2004 */
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
STATUS
approved