A334467 - OEIS

%I #20 Nov 30 2020 16:23:12

%S 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 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 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 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 T(n,k) = n*A323345(n,k).

%e Array begins:

%e      k  0   1   2   3   4   5   6   7   8   9  10

%e    n +------------------------------------------------

%e    1 |  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1, ...

%e    2 |  4,  2,  2,  2,  2,  2,  2,  2,  2,  2,  2, ...

%e    3 |  6,  6,  3,  3,  3,  3,  3,  3,  3,  3,  3, ...

%e    4 | 12,  4,  8,  4,  4,  4,  4,  4,  4,  4,  4, ...

%e    5 | 10, 10,  5, 10,  5,  5,  5,  5,  5,  5,  5, ...

%e    6 | 24, 12, 12,  6, 12,  6,  6,  6,  6,  6,  6, ...

%e    7 | 14, 14,  7, 14,  7, 14,  7,  7,  7,  7,  7, ...

%e    8 | 32,  8, 16,  8, 16,  8, 16,  8,  8,  8,  8, ...

%e    9 | 27, 27, 18, 18,  9, 18,  9, 18,  9,  9,  9, ...

%e   10 | 40, 20, 20, 10, 20, 20, 20, 10, 20, 10, 10, ...

%e ...

%t nmax = 13;

%t 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 T[n_, k_] := n col[k][[n]];

%t Table[T[n-k, k], {n, 1, nmax}, {k, 0, n-1}] // Flatten (* _Jean-François Alcover_, Nov 30 2020 *)

%Y Columns k: A038040 (k=0), A245579 (k=1), A060872 (k=2), A334463 (k=3), A327262 (k=4), A334733 (k=5), A334953 (k=6).

%Y Every diagonal starting with 1 gives A000027.

%Y 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 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 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 Cf. A245579, A323345, A334466.

%K nonn,tabl

%O 1,2

%A _Omar E. Pol_, May 05 2020