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A060642 Triangle read by rows: row n lists number of ordered partitions into k parts of partitions of n. 22
1, 2, 1, 3, 4, 1, 5, 10, 6, 1, 7, 22, 21, 8, 1, 11, 43, 59, 36, 10, 1, 15, 80, 144, 124, 55, 12, 1, 22, 141, 321, 362, 225, 78, 14, 1, 30, 240, 669, 944, 765, 370, 105, 16, 1, 42, 397, 1323, 2266, 2287, 1437, 567, 136, 18, 1, 56, 640, 2511, 5100, 6215, 4848, 2478, 824, 171, 20, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Also the convolution triangle of A000041. - Peter Luschny, Oct 07 2022
LINKS
FORMULA
G.f. A(n;x) for n-th row satisfies A(n;x) = Sum_{k=0..n-1} A000041(n-k)*A(k;x)*x, A(0;x) = 1. - Vladeta Jovovic, Jan 02 2004
T(n,k) = Sum_{i=0..k} (-1)^i * C(k,i) * A144064(n,k-i). - Alois P. Heinz, Mar 12 2015
Sum_{k=1..n} k * T(n,k) = A326346(n). - Alois P. Heinz, Sep 11 2019
Sum_{k=0..n} (-1)^k * T(n,k) = A010815(n). - Alois P. Heinz, Feb 07 2021
G.f. of column k: (-1 + Product_{j>=1} 1 / (1 - x^j))^k. - Ilya Gutkovskiy, Feb 13 2021
EXAMPLE
Table begins:
1;
2, 1;
3, 4, 1;
5, 10, 6, 1;
7, 22, 21, 8, 1;
11, 43, 59, 36, 10, 1;
15, 80, 144, 124, 55, 12, 1;
22, 141, 321, 362, 225, 78, 14, 1;
30, 240, 669, 944, 765, 370, 105, 16, 1;
42, 397, 1323, 2266, 2287, 1437, 567, 136, 18, 1;
...
For n=4 there are 5 partitions of 4, namely 4, 31, 22, 211, 11111. There are 5 ways to pick 1 of them; 10 ways to partition one of them into 2 ordered parts: 3,1; 1,3; 2,2; 21,1; 1,21; 2,11; 11,2; 111,1; 1,111; 11,11; 6 ways to partition one of them into 3 ordered parts: 2,1,1; 1,2,1; 1,1,2; 11,1,1; 1,11,1; 1,1,11; and one way to partition one of them into 4 ordered parts: 1,1,1,1. So row 4 is 5,10,6,1.
MAPLE
A:= proc(n, k) option remember; `if`(n=0, 1, k*add(
A(n-j, k)*numtheory[sigma](j), j=1..n)/n)
end:
T:= (n, k)-> add(A(n, k-i)*(-1)^i*binomial(k, i), i=0..k):
seq(seq(T(n, k), k=1..n), n=1..12); # Alois P. Heinz, Mar 12 2015
# Uses function PMatrix from A357368. Adds row and column for n, k = 0.
PMatrix(10, combinat:-numbpart); # Peter Luschny, Oct 07 2022
MATHEMATICA
A[n_, k_] := A[n, k] = If[n==0, 1, k*Sum[A[n-j, k]*DivisorSigma[1, j], {j, 1, n}]/n]; T[n_, k_] := Sum[A[n, k-i]*(-1)^i*Binomial[k, i], {i, 0, k}]; Table[ Table[ T[n, k], {k, 1, n}], {n, 1, 12}] // Flatten (* Jean-François Alcover, Jul 15 2015, after Alois P. Heinz *)
CROSSREFS
Row sums give A055887.
T(2n,n) gives A340987.
Sequence in context: A126198 A055888 A094442 * A306186 A154929 A249042
KEYWORD
easy,nonn,tabl
AUTHOR
Alford Arnold, Apr 16 2001
EXTENSIONS
More terms from Vladeta Jovovic, Jan 02 2004
STATUS
approved

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Last modified March 28 10:31 EDT 2024. Contains 371240 sequences. (Running on oeis4.)