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%I #18 Aug 16 2020 20:50:30
%S 0,0,0,0,0,0,1,1,1,1,1,1,2,4,4,4,4,4,5,7,10,11,11,11,12,14,18,23,25,
%T 26,27,29,33,40,47,52,55,58,62,70,81,93,102,109,115,124,137,155,173,
%U 190,203,216,232,255,283,313,340,365,388,417,454,499,544,590,631,674
%N Number of partitions of n into parts 7k+1 and 7k+6 with at least one part of each type.
%H Robert Price, <a href="/A035661/b035661.txt">Table of n, a(n) for n = 1..1000</a>
%F G.f. : (-1 + 1/Product_{k>=0} (1 - x^(7 k + 6)))*(-1 + 1/Product_{k>=0} (1 - x^(7 k + 1))). - _Robert Price_, Aug 14 2020
%t nmax = 66; s1 = Range[0, nmax/7]*7 + 1; s2 = Range[0, nmax/7]*7 + 6;
%t Table[Count[IntegerPartitions[n, All, s1~Join~s2],
%t x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 1, nmax}] (* _Robert Price_, Aug 14 2020 *)
%t nmax = 66; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(7 k + 1)), {k, 0, nmax}])*(-1 + 1/Product[(1 - x^(7 k + 6)), {k, 0, nmax}]), {x, 0, nmax}], x] (* _Robert Price_, Aug 14 2020 *)
%Y Cf. A035441-A035468, A035618-A035660, A035662-A035699.
%K nonn
%O 1,13
%A _Olivier Gérard_
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