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%I #7 Apr 21 2018 20:57:48
%S 1,0,1,1,0,1,1,0,0,1,2,0,0,0,1,2,1,0,0,0,1,4,0,0,0,0,0,1,4,1,0,0,0,0,
%T 0,1,6,0,1,0,0,0,0,0,1,7,2,0,0,0,0,0,0,0,1,11,0,0,0,0,0,0,0,0,0,1,10,
%U 2,1,1,0,0,0,0,0,0,0,1,17,0,0,0,0,0,0,0,0,0,0,0,1,17,4,0,0,0,0,0,0,0,0,0,0,0,1,23,0,2,0,1
%N Regular triangle where T(n,k) is the number of strict integer partitions of n with greatest common divisor k.
%F If k divides n, T(n,k) = A078374(n/k); otherwise T(n,k) = 0.
%e Triangle begins:
%e 01: 1
%e 02: 0 1
%e 03: 1 0 1
%e 04: 1 0 0 1
%e 05: 2 0 0 0 1
%e 06: 2 1 0 0 0 1
%e 07: 4 0 0 0 0 0 1
%e 08: 4 1 0 0 0 0 0 1
%e 09: 6 0 1 0 0 0 0 0 1
%e 10: 7 2 0 0 0 0 0 0 0 1
%e 11: 11 0 0 0 0 0 0 0 0 0 1
%e 12: 10 2 1 1 0 0 0 0 0 0 0 1
%e 13: 17 0 0 0 0 0 0 0 0 0 0 0 1
%e 14: 17 4 0 0 0 0 0 0 0 0 0 0 0 1
%e 15: 23 0 2 0 1 0 0 0 0 0 0 0 0 0 1
%e The strict partitions counted in row 12 are the following.
%e T(12,1) = 10: (11,1) (9,2,1) (8,3,1) (7,5) (7,4,1) (7,3,2) (6,5,1) (6,3,2,1) (5,4,3) (5,4,2,1)
%e T(12,2) = 2: (10,2) (6,4,2)
%e T(12,3) = 1: (9,3)
%e T(12,4) = 1: (8,4)
%e T(12,12) = 1: (12)
%t Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&GCD@@#===k&]],{n,15},{k,n}]
%Y First column is A078374. Second column at even indices is same as first column. Row sums are A000009. Row sums with first column removed are A303280.
%Y Cf. A000009, A000837, A018783, A051424, A117408, A168532, A289508, A289509, A298748, A300486, A302698, A302796, A303139, A303140, A303280.
%K nonn,tabl
%O 1,11
%A _Gus Wiseman_, Apr 19 2018
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