A060642 Triangle read by rows: row n lists number of ordered partitions into k parts of partitions of n.
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
Offset: 1
Examples
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.
Links
- Alois P. Heinz, Rows n = 1..141, flattened
Crossrefs
Programs
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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 *)
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
Extensions
More terms from Vladeta Jovovic, Jan 02 2004
Comments