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%I #17 Aug 17 2020 07:29:09
%S 0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,10,11,11,11,11,11,11,
%T 11,22,25,26,26,26,26,26,26,44,51,54,55,55,55,55,55,84,98,105,108,109,
%U 109,109,109,153,178,193,200,203,204,204,204,270,313,341,356,363,366
%N Number of partitions of n into parts 8k and 8k+1 with at least one part of each type.
%H Robert Price, <a href="/A035672/b035672.txt">Table of n, a(n) for n = 1..1000</a>
%F G.f.: (-1 + 1/Product_{k>=0} (1 - x^(8*k + 1)))*(-1 + 1/Product_{k>=1} (1 - x^(8*k))). - _Robert Price_, Aug 12 2020
%t nmax = 70; s1 = Range[1, nmax/8]*8; s2 = Range[0, nmax/8]*8 + 1;
%t Table[Count[IntegerPartitions[n, All, s1~Join~s2],
%t x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 1, nmax}] (* _Robert Price_, Aug 12 2020 *)
%t nmax = 70; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(8 k)), {k, 1, nmax}])*(-1 + 1/Product[(1 - x^(8 k + 1)), {k, 0, nmax}]), {x, 0, nmax}], x] (* _Robert Price_, Aug 12 2020 *)
%Y Cf. A035441-A035468, A035618-A035671, A035673-A035699.
%K nonn
%O 1,17
%A _Olivier Gérard_
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