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A328362 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).

%I A328362 #37 Oct 23 2019 08:56:09
%S A328362 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 A328362 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 A328362 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 A328362 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 A328362 Iff n is a power of 2 (A000079) then row n lists n - 1 zeros together with n.
%C A328362 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 A328362 T(n,k) = k*A328361(n,k).
%e A328362 Triangle begins:
%e A328362 1;
%e A328362 0, 2;
%e A328362 1, 2, 3;
%e A328362 0, 0, 0, 4;
%e A328362 0, 2, 3, 0, 5;
%e A328362 1, 2, 3, 0, 0, 6;
%e A328362 0, 0, 3, 4, 0, 0, 7;
%e A328362 0, 0, 0, 0, 0, 0, 0, 8;
%e A328362 0, 2, 3, 8, 5, 0, 0, 0, 9;
%e A328362 1, 2, 3, 4, 0, 0, 0, 0, 0, 10;
%e A328362 0, 0, 0, 0, 5, 6, 0, 0, 0,  0, 11;
%e A328362 0, 0, 3, 4, 5, 0, 0, 0, 0,  0,  0, 12;
%e A328362 0, 0, 0, 0, 0, 6, 7, 0, 0,  0,  0,  0, 13;
%e A328362 0, 2, 3, 4, 5, 0, 0, 0, 0,  0,  0,  0,  0, 14;
%e A328362 1, 2, 3, 8,10, 6, 7, 8, 0,  0,  0,  0,  0,  0, 15;
%e A328362 0, 0, 0, 0, 0, 0, 0, 0, 0,  0,  0,  0,  0,  0,  0, 16;
%e A328362 ...
%e A328362 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 A328362 Row sums give A245579.
%Y A328362 Column 1 gives A010054, n => 1.
%Y A328362 Leading diagonal gives A000027.
%Y A328362 Cf. A000079, A001227, A065091, A138785, A204217, A237048, A237593, A266531, A285898, A285899, A285900, A285914, A286000, A286001, A299765, A328361.
%K A328362 nonn,tabl
%O A328362 1,3
%A A328362 _Omar E. Pol_, Oct 20 2019