A334466 Square array read by antidiagonals upwards: T(n,k) is the total number of parts in all partitions of n into consecutive parts that differ by k, with n >= 1, k >= 0.
1, 3, 1, 4, 1, 1, 7, 3, 1, 1, 6, 1, 1, 1, 1, 12, 3, 3, 1, 1, 1, 8, 4, 1, 1, 1, 1, 1, 15, 3, 3, 3, 1, 1, 1, 1, 13, 1, 1, 1, 1, 1, 1, 1, 1, 18, 6, 3, 3, 3, 1, 1, 1, 1, 1, 12, 5, 4, 1, 1, 1, 1, 1, 1, 1, 1, 28, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 14, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 24, 3, 6, 3, 3, 3, 3, 1
Offset: 1
Examples
Square array starts: n\k| 0 1 2 3 4 5 6 7 8 9 10 11 12 ---+--------------------------------------------- 1 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ... 2 | 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ... 3 | 4, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ... 4 | 7, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ... 5 | 6, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, ... 6 | 12, 4, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, ... 7 | 8, 3, 1, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, ... 8 | 15, 1, 3, 1, 3, 1, 3, 1, 1, 1, 1, 1, 1, ... 9 | 13, 6, 4, 3, 1, 3, 1, 3, 1, 1, 1, 1, 1, ... 10 | 18, 5. 3. 1. 3. 1, 3, 1, 3, 1, 1, 1, 1, ... 11 | 12, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 1, 1, ... 12 | 28, 4, 6, 4, 3, 1, 3, 1, 3, 1, 3, 1, 1, ... ... For n = 9 we have that: For k = 0 the partitions of 9 into consecutive parts that differ by 0 (or simply: the partitions of 9 into equal parts) are [9], [3,3,3], [1,1,1,1,1,1,1,1,1]. In total there are 13 parts, so T(9,0) = 13. For k = 1 the partitions of 9 into consecutive parts that differ by 1 (or simply: the partitions of 9 into consecutive parts) are [9], [5,4], [4,3,2]. In total there are six parts, so T(9,1) = 6. For k = 2 the partitions of 9 into consecutive parts that differ by 2 are [9], [5, 3, 1]. In total there are four parts, so T(9,2) = 4.
Crossrefs
Columns k: A000203 (k=0), A204217 (k=1), A066839 (k=2), A330889 (k=3), A334464 (k=4), A334732 (k=5), A334949 (k=6), A377300 (k=7), A377301 (k=8).
Triangles whose row sums give the column k: A127093 (k=0), A285914 (k=1), A330466 (k=2) (conjectured), A330888 (k=3), A334462 (k=4), A334540 (k=5), A339947 (k=6).
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).
Tables of partitions related to column k: A010766 (k=0), A286001 (k=1), A332266 (k=2), A334945 (k=3), A334618 (k=4).
Programs
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Mathematica
nmax = 14; col[k_] := col[k] = CoefficientList[Sum[n x^(n(k n - k + 2)/2)/(1 - x^n), {n, 1, nmax}] + O[x]^(nmax+1), x]; T[n_, k_] := col[k][[n+1]]; Table[T[n-k, k], {n, 1, nmax}, {k, 0, n-1}] // Flatten (* Jean-François Alcover, Nov 30 2020 *)
Formula
The g.f. for column k is Sum_{n>=1} n*x^(n*(k*n-k+2)/2)/(1-x^n). (For proof, see A330889. - N. J. A. Sloane, Nov 21 2020)
Comments