A036021 - OEIS

%I #12 May 09 2018 10:07:00

%S 1,1,1,1,2,3,3,4,6,8,9,11,15,18,21,26,33,40,46,55,68,81,94,111,134,

%T 158,182,213,252,294,338,392,459,531,607,699,810,930,1059,1212,1393,

%U 1590,1804,2052,2342,2660,3005,3403,3862,4365,4914,5540,6255,7040,7899,8871

%N Number of partitions of n into parts not of form 4k+2, 16k, 16k+3 or 16k-3.

%C Case k=4,i=2 of Gordon/Goellnitz/Andrews Theorem.

%C Also number of partitions in which no odd part is repeated, with 1 part 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.

%D G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 114.

%F Euler transform of period 16 sequence [1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 0, ...]. - _Michael Somos_, Jul 15 2004

%F 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

%o (PARI) a(n)=if(n<0,0,polcoeff(1/prod(k=1,n,1-([1,0,0,1,1,0,1,1,1,0,1,1,0,0,1,0][(k-1)%16+1])*x^k,1+x*O(x^n)),n))

%K nonn,easy

%O 0,5

%A _Olivier Gérard_