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A334467 Square array read by antidiagonals upwards: T(n,k) is the sum of all parts of all partitions of n into consecutive parts that differ by k, with n >= 1, k >= 0.

%I A334467 #20 Nov 30 2020 16:23:12
%S A334467 1,4,1,6,2,1,12,6,2,1,10,4,3,2,1,24,10,8,3,2,1,14,12,5,4,3,2,1,32,14,
%T A334467 12,10,4,3,2,1,27,8,7,6,5,4,3,2,1,40,27,16,14,12,5,4,3,2,1,22,20,18,8,
%U A334467 7,6,5,4,3,2,1,72,22,20,18,16,14,6,5,4,3,2,1,26,24,11,10,9,8,7,6,5,4,3,2,1
%N A334467 Square array read by antidiagonals upwards: T(n,k) is the sum of all parts of all partitions of n into consecutive parts that differ by k, with n >= 1, k >= 0.
%F A334467 T(n,k) = n*A323345(n,k).
%e A334467 Array begins:
%e A334467      k  0   1   2   3   4   5   6   7   8   9  10
%e A334467    n +------------------------------------------------
%e A334467    1 |  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1, ...
%e A334467    2 |  4,  2,  2,  2,  2,  2,  2,  2,  2,  2,  2, ...
%e A334467    3 |  6,  6,  3,  3,  3,  3,  3,  3,  3,  3,  3, ...
%e A334467    4 | 12,  4,  8,  4,  4,  4,  4,  4,  4,  4,  4, ...
%e A334467    5 | 10, 10,  5, 10,  5,  5,  5,  5,  5,  5,  5, ...
%e A334467    6 | 24, 12, 12,  6, 12,  6,  6,  6,  6,  6,  6, ...
%e A334467    7 | 14, 14,  7, 14,  7, 14,  7,  7,  7,  7,  7, ...
%e A334467    8 | 32,  8, 16,  8, 16,  8, 16,  8,  8,  8,  8, ...
%e A334467    9 | 27, 27, 18, 18,  9, 18,  9, 18,  9,  9,  9, ...
%e A334467   10 | 40, 20, 20, 10, 20, 20, 20, 10, 20, 10, 10, ...
%e A334467 ...
%t A334467 nmax = 13;
%t A334467 col[k_] := col[k] = CoefficientList[Sum[x^(n(k n - k + 2)/2 - 1)/(1 - x^n), {n, 1, nmax}] + O[x]^nmax, x];
%t A334467 T[n_, k_] := n col[k][[n]];
%t A334467 Table[T[n-k, k], {n, 1, nmax}, {k, 0, n-1}] // Flatten (* _Jean-François Alcover_, Nov 30 2020 *)
%Y A334467 Columns k: A038040 (k=0), A245579 (k=1), A060872 (k=2), A334463 (k=3), A327262 (k=4), A334733 (k=5), A334953 (k=6).
%Y A334467 Every diagonal starting with 1 gives A000027.
%Y A334467 Sequences of number of parts related to column k: A000203 (k=0), A204217 (k=1), A066839 (k=2) (conjectured), A330889 (k=3), A334464 (k=4), A334732 (k=5), A334949 (k=6).
%Y A334467 Sequences of number of partitions related to column k: A000005 (k=0), A001227 (k=1), A038548 (k=2), A117277 (k=3), A334461 (k=4), A334541 (k=5), A334948 (k=6).
%Y A334467 Polygonal  numbers related to column k: A001477 (k=0), A000217 (k=1), A000290 (k=2), A000326 (k=3), A000384 (k=4), A000566 (k=5), A000567 (k=6).
%Y A334467 Cf. A245579, A323345, A334466.
%K A334467 nonn,tabl
%O A334467 1,2
%A A334467 _Omar E. Pol_, May 05 2020