OFFSET
0,4
COMMENTS
Case k=4,i=4 of Gordon/Goellnitz/Andrews Theorem.
Also number of partitions in which no odd part is repeated, with at most 3 parts of size less than or equal to 2 and where differences between parts at distance 3 are greater than 1 when the larger part is odd and greater than 2 when the larger part is even.
Euler transform of period 16 sequence [1,0,1,1,1,0,0,1,0,0,1,1,1,0,1,0,...]. - Michael Somos, Jul 15 2004
REFERENCES
G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 114.
FORMULA
a(n) ~ 3^(1/4) * sqrt(2 + sqrt(2 + sqrt(2))) * exp(sqrt(3*n/2)*Pi/2) / (2^(15/4) * n^(3/4)). - Vaclav Kotesovec, May 09 2018
MATHEMATICA
a[n_] := If[n < 0, 0, SeriesCoefficient[ 1/Product[ 1 - ({1, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0}[[Mod[k - 1, 16] + 1]])*x^k, {k, 1, n}], {x, 0, n}]]; Table[a[n], {n, 0, 52}] (* Jean-François Alcover, Jul 19 2013, translated from Pari *)
PROG
(PARI) a(n)=if(n<0, 0, polcoeff(1/prod(k=1, n, 1-([1, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0][(k-1)%16+1])*x^k, 1+x*O(x^n)), n))
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
STATUS
approved