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A221530
Triangle read by rows: T(n,k) = A000005(k)*A000041(n-k).
17
1, 1, 2, 2, 2, 2, 3, 4, 2, 3, 5, 6, 4, 3, 2, 7, 10, 6, 6, 2, 4, 11, 14, 10, 9, 4, 4, 2, 15, 22, 14, 15, 6, 8, 2, 4, 22, 30, 22, 21, 10, 12, 4, 4, 3, 30, 44, 30, 33, 14, 20, 6, 8, 3, 4, 42, 60, 44, 45, 22, 28, 10, 12, 6, 4, 2, 56, 84, 60, 66, 30, 44, 14, 20, 9, 8, 2, 6
OFFSET
1,3
COMMENTS
T(n,k) is the number of partitions of n that contain k as a part multiplied by the number of divisors of k.
It appears that T(n,k) is also the total number of appearances of k in the last k sections of the set of partitions of n multiplied by the number of divisors of k.
T(n,k) is also the number of partitions of k into equal parts multiplied by the number of ones in the j-th section of the set of partitions of n, where j = (n - k + 1).
For another version see A245095. - Omar E. Pol, Jul 15 2014
LINKS
Paolo Xausa, Table of n, a(n) for n = 1..11325 (rows 1..150 of the triangle, flattened)
FORMULA
T(n,k) = d(k)*p(n-k) = A000005(k)*A027293(n,k).
EXAMPLE
For n = 6:
-------------------------
k A000005 T(6,k)
1 1 * 7 = 7
2 2 * 5 = 10
3 2 * 3 = 6
4 3 * 2 = 6
5 2 * 1 = 2
6 4 * 1 = 4
-------------------------
So row 6 is [7, 10, 6, 6, 4, 2]. Note that the sum of row 6 is 7+10+6+6+2+4 = 35 equals A006128(6).
.
Triangle begins:
1;
1, 2;
2, 2, 2;
3, 4, 2, 3;
5, 6, 4, 3, 2;
7, 10, 6, 6, 2, 4;
11, 14, 10, 9, 4, 4, 2;
15, 22, 14, 15, 6, 8, 2, 4;
22, 30, 22, 21, 10, 12, 4, 4, 3;
30, 44, 30, 33, 14, 20, 6, 8, 3, 4;
42, 60, 44, 45, 22, 28, 10, 12, 6, 4, 2;
56, 84, 60, 66, 30, 44, 14, 20, 9, 8, 2, 6;
...
MATHEMATICA
A221530row[n_]:=DivisorSigma[0, Range[n]]PartitionsP[n-Range[n]]; Array[A221530row, 10] (* Paolo Xausa, Sep 04 2023 *)
PROG
(PARI) row(n) = vector(n, i, numdiv(i)*numbpart(n-i)); \\ Michel Marcus, Jul 18 2014
CROSSREFS
Similar to A221529.
Columns 1-2: A000041, A139582. Leading diagonals 1-3: A000005, A000005, A062011. Row sums give A006128.
Sequence in context: A216644 A187758 A171931 * A262944 A322789 A069904
KEYWORD
nonn,tabl
AUTHOR
Omar E. Pol, Jan 19 2013
STATUS
approved