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%I #37 Oct 23 2019 08:56:09
%S 1,0,2,1,2,3,0,0,0,4,0,2,3,0,5,1,2,3,0,0,6,0,0,3,4,0,0,7,0,0,0,0,0,0,
%T 0,8,0,2,3,8,5,0,0,0,9,1,2,3,4,0,0,0,0,0,10,0,0,0,0,5,6,0,0,0,0,11,0,
%U 0,3,4,5,0,0,0,0,0,0,12,0,0,0,0,0,6,7,0,0,0,0,0,13,0,2,3,4,5,0,0,0,0,0,0,0,0,14
%N Triangle read by rows: T(n,k) is the sum of all parts k in all partitions of n into consecutive parts, (1 <= k <= n).
%C Iff n is a power of 2 (A000079) then row n lists n - 1 zeros together with n.
%C Iff n is an odd prime (A065091) then row n lists (n - 3)/2 zeros, (n - 1)/2, (n + 1)/2, (n - 3)/2 zeros, n.
%F T(n,k) = k*A328361(n,k).
%e Triangle begins:
%e 1;
%e 0, 2;
%e 1, 2, 3;
%e 0, 0, 0, 4;
%e 0, 2, 3, 0, 5;
%e 1, 2, 3, 0, 0, 6;
%e 0, 0, 3, 4, 0, 0, 7;
%e 0, 0, 0, 0, 0, 0, 0, 8;
%e 0, 2, 3, 8, 5, 0, 0, 0, 9;
%e 1, 2, 3, 4, 0, 0, 0, 0, 0, 10;
%e 0, 0, 0, 0, 5, 6, 0, 0, 0, 0, 11;
%e 0, 0, 3, 4, 5, 0, 0, 0, 0, 0, 0, 12;
%e 0, 0, 0, 0, 0, 6, 7, 0, 0, 0, 0, 0, 13;
%e 0, 2, 3, 4, 5, 0, 0, 0, 0, 0, 0, 0, 0, 14;
%e 1, 2, 3, 8,10, 6, 7, 8, 0, 0, 0, 0, 0, 0, 15;
%e 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 16;
%e ...
%e For n = 9 there are three partitions of 9 into consecutive parts, they are [9], [5, 4], [4, 3, 2], so the 9th row of triangle is [0, 2, 3, 8, 5, 0, 0, 0, 9].
%Y Row sums give A245579.
%Y Column 1 gives A010054, n => 1.
%Y Leading diagonal gives A000027.
%Y Cf. A000079, A001227, A065091, A138785, A204217, A237048, A237593, A266531, A285898, A285899, A285900, A285914, A286000, A286001, A299765, A328361.
%K nonn,tabl
%O 1,3
%A _Omar E. Pol_, Oct 20 2019
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