OFFSET
0,5
COMMENTS
In each row n>=0, k takes values from 0 to n^2 inclusive. Row sums equal A000169(n+1). All rows are palindromic. Row n is also row n of the (n+1)-nomial array (e.g., row 1 is also row 1 of A007318).
T(n,k) gives the number of divisors of A181555(n) with k prime factors counted with multiplicity. See also A001222, A071207, A146291, A146292.
T(n,k) is the number of size k submultisets of the so-called regular multiset {1_1,1_2,...,1_(n-1),1_n, ... ,i_1,i_2,...,i_(n-1),i_n, ... ,n_1,n_2,...,n_(n-1),n_n} (which contains n copies of i for 0 < i < n). - Thomas Wieder, Dec 28 2013
LINKS
Alois P. Heinz, Rows n = 0..32, flattened
Anonymous, Polynomial calculator
Thomas Wieder, A181567 as Excel table
G. Xiao, WIMS server, Factoris (both expands and factors polynomials)
EXAMPLE
Rows begin:
1;
1,1;
1,2,3,2,1;
1,3,6,10,12,12,10,6,3,1;...
T(n=3,k=4) = 12 because we have 12 submultisets (without regard of the order of elements) of size k=4 for the regular multiset (n=3) {1, 1, 1, 2, 2, 2, 3, 3, 3}: {1, 1, 1, 2}, {1, 1, 1, 3}, {1, 1, 2, 2}, {1, 1, 2, 3}, {1, 1, 3, 3}, {1, 2, 2, 2}, {1, 2, 2, 3}, {1, 2, 3, 3}, {1, 3, 3, 3}, {2, 2, 2, 3}, {2, 2, 3, 3}, {2, 3, 3, 3}.
MAPLE
b:= proc(n, k, i) option remember; `if`(k=0, 1,
`if`(i<1, 0, add(b(n, k-j, i-1), j=0..n)))
end:
T:= (n, k)-> b(n, k, n):
seq(seq(T(n, k), k=0..n^2), n=0..8); # Alois P. Heinz, Jul 04 2016
MATHEMATICA
row[n_] := CoefficientList[((x^(n+1) - 1)/(x-1))^n + O[x]^(n^2+1), x]; Table[row[n], {n, 0, 6}] // Flatten (* Jean-François Alcover, Apr 06 2017 *)
CROSSREFS
KEYWORD
easy,nonn,tabf
AUTHOR
Matthew Vandermast, Oct 31 2010
STATUS
approved