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A334467
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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.
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1
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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, 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, 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
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OFFSET
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1,2
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LINKS
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Table of n, a(n) for n=1..91.
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FORMULA
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T(n,k) = n*A323345(n,k).
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EXAMPLE
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Array begins:
k 0 1 2 3 4 5 6 7 8 9 10
n +------------------------------------------------
1 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
2 | 4, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, ...
3 | 6, 6, 3, 3, 3, 3, 3, 3, 3, 3, 3, ...
4 | 12, 4, 8, 4, 4, 4, 4, 4, 4, 4, 4, ...
5 | 10, 10, 5, 10, 5, 5, 5, 5, 5, 5, 5, ...
6 | 24, 12, 12, 6, 12, 6, 6, 6, 6, 6, 6, ...
7 | 14, 14, 7, 14, 7, 14, 7, 7, 7, 7, 7, ...
8 | 32, 8, 16, 8, 16, 8, 16, 8, 8, 8, 8, ...
9 | 27, 27, 18, 18, 9, 18, 9, 18, 9, 9, 9, ...
10 | 40, 20, 20, 10, 20, 20, 20, 10, 20, 10, 10, ...
...
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MATHEMATICA
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nmax = 13;
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[n_, k_] := n col[k][[n]];
Table[T[n-k, k], {n, 1, nmax}, {k, 0, n-1}] // Flatten (* Jean-François Alcover, Nov 30 2020 *)
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CROSSREFS
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Columns k: A038040 (k=0), A245579 (k=1), A060872 (k=2), A334463 (k=3), A327262 (k=4), A334733 (k=5), A334953 (k=6).
Every diagonal starting with 1 gives A000027.
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).
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).
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).
Cf. A245579, A323345, A334466.
Sequence in context: A032145 A032050 A109915 * A255198 A202521 A247362
Adjacent sequences: A334464 A334465 A334466 * A334468 A334469 A334470
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KEYWORD
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nonn,tabl
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AUTHOR
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Omar E. Pol, May 05 2020
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STATUS
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approved
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