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A373272
Triangle read by rows: T(n,k) = sum of all distinct multiplicities in the integer partitions of n with k parts.
3
1, 1, 2, 1, 1, 3, 1, 3, 3, 4, 1, 2, 6, 4, 5, 1, 4, 7, 6, 5, 6, 1, 3, 10, 11, 10, 6, 7, 1, 5, 11, 16, 14, 12, 7, 8, 1, 4, 15, 20, 22, 14, 14, 8, 9, 1, 6, 16, 26, 28, 29, 20, 16, 9, 10, 1, 5, 20, 34, 41, 40, 34, 23, 18, 10, 11, 1, 7, 22, 42, 50, 54, 44, 35, 26, 20, 11, 12, 1, 6, 26, 52, 69, 75, 68, 54, 44, 29, 22, 12, 13
OFFSET
1,3
LINKS
Alois P. Heinz, Rows n = 1..200, flattened (first 40 rows from Olivier Gérard)
EXAMPLE
Array begins:
1;
1, 2;
1, 1, 3;
1, 3, 3, 4;
1, 2, 6, 4, 5;
1, 4, 7, 6, 5, 6;
1, 3, 10, 11, 10, 6, 7;
1, 5, 11, 16, 14, 12, 7, 8;
1, 4, 15, 20, 22, 14, 14, 8, 9;
1, 6, 16, 26, 28, 29, 20, 16, 9, 10;
...
T(6,3) = 7 because the partitions of 6 into 3 parts are 4+1+1, 3+2+1, 2+2+2,
the multiplicities are (1,2), (1,1,1), (3),
the distinct multiplicities are respectively (1,2), (1), (3),
contributing 3+1+3 = 7.
MATHEMATICA
Flatten[Table[
Plus @@@
Table[Map[Plus @@ Union[Length /@ Split[#]] &,
IntegerPartitions[n, {k}]], {k, 1, n}], {n, 1, 20}]]
CROSSREFS
Columns k=1-2 give: A057427, A028242.
Main diagonal gives A000027.
Row sums are A373273.
T(2n,n) gives A373104.
Sequence in context: A136405 A210871 A308399 * A287601 A035667 A092865
KEYWORD
nonn,tabl
AUTHOR
Olivier Gérard, May 29 2024
STATUS
approved