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A359279 Irregular triangle T(n,k) (n>=1, k>=1) read by rows in which the length of row n equals the partition number A000041(n-1) and every column k gives the positive triangular numbers A000217.

%I A359279 #34 Mar 31 2023 14:20:07
%S A359279 1,3,6,1,10,3,1,15,6,3,1,1,21,10,6,3,3,1,1,28,15,10,6,6,3,3,1,1,1,1,
%T A359279 36,21,15,10,10,6,6,3,3,3,3,1,1,1,1,45,28,21,15,15,10,10,6,6,6,6,3,3,
%U A359279 3,3,1,1,1,1,1,1,1,55,36,28,21,21,15,15,10,10,10,10,6,6,6,6,3,3,3,3,3,3,3,1,1,1,1,1,1,1,1
%N A359279 Irregular triangle T(n,k) (n>=1, k>=1) read by rows in which the length of row n equals the partition number A000041(n-1) and every column k gives the positive triangular numbers A000217.
%C A359279 All divisors of the largest partition into consecutive parts of all terms in row n are also all parts of all partitions of n.
%H A359279 Paolo Xausa, <a href="/A359279/b359279.txt">Table of n, a(n) for n = 1..11732</a> (rows 1..27 of triangle, flattened).
%F A359279 T(n,k) = A000217(A336811(n,k)).
%e A359279 Triangle begins:
%e A359279    1;
%e A359279    3;
%e A359279    6,  1;
%e A359279   10,  3,  1;
%e A359279   15,  6,  3,  1,  1;
%e A359279   21, 10,  6,  3,  3,  1,  1;
%e A359279   28, 15, 10,  6,  6,  3,  3, 1, 1, 1, 1;
%e A359279   36, 21, 15, 10, 10,  6,  6, 3, 3, 3, 3, 1, 1, 1, 1;
%e A359279   45, 28, 21, 15, 15, 10, 10, 6, 6, 6, 6, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1;
%e A359279 ...
%e A359279 From _Omar E. Pol_, Feb 28 2023: (Start)
%e A359279 For n = 4 the fourth row is [10, 3, 1]. The largest partition into consecutive parts of every term are respectively [4, 3, 2, 1], [2, 1], [1]. The divisors of these parts are [(1, 2, 4), (1, 3), (1, 2), (1)], [(1, 2), (1)], [1]. These 12 divisors are also all parts of all partitions of 4. They are  [(4), (2, 2), (3, 1), (2, 1, 1), (1, 1, 1, 1)]. (End)
%t A359279 A359279[rowmax_]:=Table[Flatten[Table[ConstantArray[(n-m)(n-m+1)/2,PartitionsP[m]-PartitionsP[m-1]],{m,0,n-1}]],{n,rowmax}];
%t A359279 A359279[10] (* Generates 10 rows *) (* _Paolo Xausa_, Mar 06 2023 *)
%o A359279 (PARI)
%o A359279 A359279(rowmax)=vector(rowmax,n,concat(vector(n,m,vector(numbpart(m-1)-numbpart(m-2),i,(n-m+1)*(n-m+2)/2))));
%o A359279 A359279(10) \\ Generates 10 rows - _Paolo Xausa_, Mar 06 2023
%Y A359279 Row sums give A014153 (convolution of A000041 and A000027).
%Y A359279 This sequence has the same row sums as A176206, A299779 and A359350.
%Y A359279 Cf. A000217, A336811, A336812.
%K A359279 nonn,tabf
%O A359279 1,2
%A A359279 _Omar E. Pol_, Dec 23 2022