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A328371 Irregular triangle read by rows: T(n,k) is the sum of all parts of all partitions of all positive integers <= n into k consecutive parts.

%I A328371 #16 Nov 05 2019 01:03:13
%S A328371 1,3,6,3,10,3,15,8,21,8,6,28,15,6,36,15,6,45,24,15,55,24,15,10,66,35,
%T A328371 15,10,78,35,27,10,91,48,27,10,105,48,27,24,120,63,42,24,15,136,63,42,
%U A328371 24,15,153,80,42,24,15,171,80,60,42,15,190,99,60,42,15,210,99,60,42,35,231,120,81,42,35,21
%N A328371 Irregular triangle read by rows: T(n,k) is the sum of all parts of all partitions of all positive integers <= n into k consecutive parts.
%C A328371 Column k lists the partial sums of the k-th column of triangle A285891.
%e A328371 Triangle begins:
%e A328371     1;
%e A328371     3;
%e A328371     6,   3;
%e A328371    10,   3;
%e A328371    15,   8;
%e A328371    21,   8,   6;
%e A328371    28,  15,   6;
%e A328371    36,  15,   6;
%e A328371    45,  24,  15;
%e A328371    55,  24,  15, 10;
%e A328371    66,  35,  15, 10;
%e A328371    78,  35,  27, 10;
%e A328371    91,  48,  27, 10;
%e A328371   105,  48,  27, 24,
%e A328371   120,  63,  42, 24, 15;
%e A328371   136,  63,  42, 24, 15;
%e A328371   153,  80,  42, 24, 15;
%e A328371   171,  80,  60, 42, 15;
%e A328371   190,  99,  60, 42, 15;
%e A328371   210,  99,  60, 42, 35;
%e A328371   231, 120,  81, 42, 35, 21;
%e A328371   253, 120,  81, 64, 35, 21;
%e A328371   276, 143,  81, 64, 35, 21;
%e A328371   300, 143, 105, 64, 35, 21;
%e A328371   325, 168, 105, 64, 60, 21;
%e A328371   351, 168, 105, 90, 60, 21;
%e A328371   378, 195, 132, 90, 60, 48;
%e A328371   406, 195, 132, 90, 60, 48, 28;
%e A328371 ...
%o A328371 (PARI) tt(n, k) = n*(if (k % 2, (n % k) == 0, ((n - k/2) % k) == 0)); \\ A285891
%o A328371 t(n, k) = sum(j=k*(k+1)/2, n, tt(j, k));
%o A328371 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 A328371 Row sums give A285900.
%Y A328371 Row n has length A003056(n).
%Y A328371 Column 1 gives the nonzero terms of A000217.
%Y A328371 Column k starts with A000217(k) in the row A000217(k).
%Y A328371 Cf. A196020, A211343, A235791, A236104, A235791, A237048, A237591, A237593, A245579, A262612, A285899, A285914, A285891, A286000, A286001, A286013, A299765, A328362, A328365, A328368.
%K A328371 nonn,tabf
%O A328371 1,2
%A A328371 _Omar E. Pol_, Nov 02 2019