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A035651 Number of partitions of n into parts 7k and 7k+1 with at least one part of each type. 3

%I #14 Aug 16 2020 20:23:35

%S 0,0,0,0,0,0,0,1,1,1,1,1,1,1,4,4,4,4,4,4,4,10,11,11,11,11,11,11,22,25,

%T 26,26,26,26,26,44,51,54,55,55,55,55,84,98,105,108,109,109,109,153,

%U 178,193,200,203,204,204,270,313,341,356,363,366,367,463,532,582,611

%N Number of partitions of n into parts 7k and 7k+1 with at least one part of each type.

%H Robert Price, <a href="/A035651/b035651.txt">Table of n, a(n) for n = 1..1000</a>

%F G.f. : (-1 + 1/Product_{k>=0} (1 - x^(7 k + 1)))*(-1 + 1/Product_{k>=1} (1 - x^(7 k))). - _Robert Price_, Aug 12 2020

%t nmax = 67; s1 = Range[1, nmax/7]*7; s2 = Range[0, nmax/7]*7 + 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 = 67; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(7 k)), {k, 1, nmax}])*(-1 + 1/Product[(1 - x^(7 k + 1)), {k, 0, nmax}]), {x, 0, nmax}], x] (* _Robert Price_, Aug 12 2020 *)

%Y Cf. A035441-A035468, A035618-A035650, A035652-A035699.

%K nonn

%O 1,15

%A _Olivier GĂ©rard_

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