|
%I #15 Aug 16 2020 16:33:58
%S 0,0,0,1,1,1,2,4,4,5,7,10,12,14,18,24,28,33,41,50,59,70,84,100,117,
%T 137,161,188,219,254,295,341,393,453,520,595,682,780,889,1011,1150,
%U 1307,1481,1673,1893,2140,2411,2713,3053,3433,3851,4313,4833,5411
%N Number of partitions of n into parts 4k+1 and 4k+3 with at least one part of each type.
%H Alois P. Heinz, <a href="/A035625/b035625.txt">Table of n, a(n) for n = 1..1000</a> (first 100 terms from Robert Price)
%F G.f.: (-1 + 1/Product_{k>=0} (1 - x^(4 k + 1)))*(-1 + 1/Product_{k>=0} (1 - x^(4 k + 3))). - _Robert Price_, Aug 16 2020
%t nmax = 54; s1 = Range[0, nmax/4]*4 + 1; s2 = Range[0, nmax/4]*4 + 3;
%t Table[Count[IntegerPartitions[n, All, s1~Join~s2],
%t x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 1, nmax}] (* _Robert Price_, Aug 06 2020 *)
%t nmax = 54; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(4 k + 3)), {k, 0, nmax}])*(-1 + 1/Product[(1 - x^(4 k + 1)), {k, 0, nmax}]), {x, 0, nmax}], x] (* _Robert Price_, Aug 16 2020 *)
%Y Cf. A035441-A035468, A035618-A035624, A035626-A035699.
%K nonn
%O 1,7
%A _Olivier Gérard_
|
