A036023 - OEIS

%I #14 May 09 2018 10:10:48

%S 1,1,1,2,3,4,5,6,9,11,13,17,22,27,32,40,50,60,71,86,105,125,146,174,

%T 209,245,285,336,396,461,534,621,725,838,963,1113,1287,1477,1689,1938,

%U 2224,2538,2888,3293,3755,4265,4830,5478,6215,7024,7923,8947,10098

%N Number of partitions of n into parts not of form 4k+2, 16k, 16k+7 or 16k-7.

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

%C 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.

%C 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

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

%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

%t 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 *)

%o (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))

%K nonn,easy

%O 0,4

%A _Olivier Gérard_