OFFSET
1,2
COMMENTS
Also the convolution triangle of A000041. - Peter Luschny, Oct 07 2022
LINKS
Alois P. Heinz, Rows n = 1..141, flattened
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
KEYWORD
AUTHOR
Alford Arnold, Apr 16 2001
EXTENSIONS
More terms from Vladeta Jovovic, Jan 02 2004
STATUS
approved