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%I #38 Oct 23 2019 08:55:55
%S 1,0,1,1,1,1,0,0,0,1,0,1,1,0,1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0,0,0,0,0,
%T 0,1,0,1,1,2,1,0,0,0,1,1,1,1,1,0,0,0,0,0,1,0,0,0,0,1,1,0,0,0,0,1,0,0,
%U 1,1,1,0,0,0,0,0,0,1,0,0,0,0,0,1,1,0,0,0,0,0,1,0,1,1,1,1,0,0,0,0,0,0,0,0,1
%N Triangle read by rows: T(n,k) is the total number of k's 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 1.
%C Iff n is an odd prime (A065091) then row n lists (n - 3)/2 zeros, 1, 1, (n - 3)/2 zeros, 1.
%e Triangle begins:
%e 1;
%e 0, 1;
%e 1, 1, 1;
%e 0, 0, 0, 1;
%e 0, 1, 1, 0, 1;
%e 1, 1, 1, 0, 0, 1;
%e 0, 0, 1, 1, 0, 0, 1;
%e 0, 0, 0, 0, 0, 0, 0, 1;
%e 0, 1, 1, 2, 1, 0, 0, 0, 1;
%e 1, 1, 1, 1, 0, 0, 0, 0, 0, 1;
%e 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1;
%e 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1;
%e 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1;
%e 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1;
%e 1, 1, 1, 2, 2, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1;
%e 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
%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, 1, 1, 2, 1, 0, 0, 0, 1].
%Y Row sums give A204217.
%Y Column 1 gives A010054, n >= 1.
%Y Leading diagonal gives A000012.
%Y Cf. A000079, A001227, A065091, A066633, A237048, A237593, A245579, A266531, A285898, A285899, A285900, A285914, A286000, A286001, A299765, A328362.
%K nonn,tabl
%O 1,40
%A _Omar E. Pol_, Oct 20 2019
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