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%I #10 Jan 11 2024 22:56:54
%S 1,1,1,1,2,1,1,3,1,2,1,4,2,3,2,1,5,2,5,3,3,1,6,3,8,4,4,4,1,7,3,11,6,6,
%T 6,5,1,8,4,14,9,8,10,7,6,1,9,4,19,11,11,14,11,9,8,1,10,5,23,14,15,21,
%U 15,14,11,10,1,11,5,28,17,19,28,22,20,17,15,12
%N Triangle read by rows where T(n,k) is the number of integer partitions with sum <= n and with distinct parts summing to k.
%C Also the number of ways to write any number up to n as a positive linear combination of a strict integer partition of k.
%H Andrew Howroyd, <a href="/A364911/b364911.txt">Table of n, a(n) for n = 0..1325</a> (rows 0..50)
%F G.f.: A(x,y) = (1/(1 - x)) * Product_{k>=1} (1 - y^k + y^k/(1 - x^k)). - _Andrew Howroyd_, Jan 11 2024
%e Triangle begins:
%e 1
%e 1 1
%e 1 2 1
%e 1 3 1 2
%e 1 4 2 3 2
%e 1 5 2 5 3 3
%e 1 6 3 8 4 4 4
%e 1 7 3 11 6 6 6 5
%e 1 8 4 14 9 8 10 7 6
%e 1 9 4 19 11 11 14 11 9 8
%e 1 10 5 23 14 15 21 15 14 11 10
%e 1 11 5 28 17 19 28 22 20 17 15 12
%e 1 12 6 34 21 22 40 28 28 24 24 17 15
%e 1 13 6 40 25 27 50 38 37 34 35 27 22 18
%e 1 14 7 46 29 32 65 49 50 43 51 38 35 26 22
%e 1 15 7 54 33 38 79 62 63 59 68 55 50 41 32 27
%e Row n = 5 counts the following partitions:
%e . 1 2 3 4 5
%e 1+1 2+2 1+2 1+3 1+4
%e 1+1+1 1+1+2 1+1+3 2+3
%e 1+1+1+1 1+1+1+2
%e 1+1+1+1+1 1+2+2
%e Row n = 5 counts the following positive linear combinations:
%e . 1*1 1*2 1*3 1*4 1*5
%e 2*1 2*2 1*2+1*1 1*3+1*1 1*3+1*2
%e 3*1 1*2+2*1 1*3+2*1 1*4+1*1
%e 4*1 1*2+3*1
%e 5*1 2*2+1*1
%t Table[Length[Select[Array[IntegerPartitions,n+1,0,Join],Total[Union[#]]==k&]],{n,0,9},{k,0,n}]
%o (PARI) T(n)={[Vecrev(p) | p<-Vec(prod(k=1, n, 1 - y^k + y^k/(1 - x^k), 1/(1 - x) + O(x*x^n)))]}
%o { my(A=T(10)); for(n=1, #A, print(A[n])) } \\ _Andrew Howroyd_, Jan 11 2024
%Y Column n = k is A000009.
%Y Column k = 0 is A000012.
%Y Column k = 1 is A000027.
%Y Row sums are A000070.
%Y Column k = 2 is A008619.
%Y Columns are partial sums of columns of A116861.
%Y Column k = 3 appears to be the partial sums of A137719.
%Y Diagonal n = 2k is A364910.
%Y A000041 counts integer partitions, strict A000009.
%Y A008284 counts partitions by length, strict A008289.
%Y A114638 counts partitions where (length) = (sum of distinct parts).
%Y A116608 counts partitions by number of distinct parts.
%Y A364350 counts combination-free strict partitions, complement A364839.
%Y Cf. A002865, A066328, A179009, A236912, A237113, A237667, A364912, A364913, A364915, A364916, A365002, A365004.
%K nonn,tabl
%O 0,5
%A _Gus Wiseman_, Aug 27 2023
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