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%I #17 Nov 04 2019 09:24:09
%S 1,2,3,2,4,2,5,4,6,4,3,7,6,3,8,6,3,9,8,6,10,8,6,4,11,10,6,4,12,10,9,4,
%T 13,12,9,4,14,12,9,8,15,14,12,8,5,16,14,12,8,5,17,16,12,8,5,18,16,15,
%U 12,5,19,18,15,12,5,20,18,15,12,10,21,20,18,12,10,6,22,20,18,16,10,6,23,22,18,16,10,6
%N Irregular triangle read by rows: T(n,k) is the total number of parts in all partitions of all positive integers <= n into k consecutive parts.
%C Column k lists k times every nonzero multiple of k in nondecreasing order.
%C Column k lists the partial sums of the k-th column of triangle A285914.
%e Triangle begins:
%e 1;
%e 2;
%e 3, 2;
%e 4, 2;
%e 5, 4;
%e 6, 4, 3;
%e 7, 6, 3;
%e 8, 6, 3;
%e 9, 8, 6;
%e 10, 8, 6, 4;
%e 11, 10, 6, 4;
%e 12, 10, 9, 4;
%e 13, 12, 9, 4;
%e 14, 12, 9, 8;
%e 15, 14, 12, 8, 5;
%e 16, 14, 12, 8, 5;
%e 17, 16, 12, 8, 5;
%e 18, 16, 15, 12, 5;
%e 19, 18, 15, 12, 5;
%e 20, 18, 15, 12, 10;
%e 21, 20, 18, 12, 10, 6;
%e 22, 20, 18, 16, 10, 6;
%e 23, 22, 18, 16, 10, 6;
%e 24, 22, 21, 16, 10, 6;
%e 25, 24, 21, 16, 15, 6;
%e 26, 24, 21, 20, 15, 6;
%e 27, 26, 24, 20, 15, 12;
%e 28, 26, 24, 20, 15, 12, 7;
%e ...
%o (PARI) tt(n, k) = k*(if (k % 2, (n % k) == 0, ((n - k/2) % k) == 0)); \\ A285891
%o t(n, k) = sum(j=k*(k+1)/2, n, tt(j, k));
%o tabf(nn) = {for (n=1, nn, for (k=1, floor((sqrt(1+8*n)-1)/2), print1(t(n, k), ", "); ); print(); ); } \\ _Michel Marcus_, Nov 04 2019
%Y Row sums give A285899.
%Y Row n has length A003056(n).
%Y Column 1 gives A000027.
%Y Column k starts with k in the row A000217(k).
%Y Cf. A052928, A196020, A204217, A211343, A235791, A236104, A235791, A237048, A237591, A237593, A245579, A262612, A285900, A285914, A285891, A286000, A286001, A286013, A299765, A328361, A328365, A328371.
%K nonn,tabf
%O 1,2
%A _Omar E. Pol_, Nov 02 2019
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