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A181567 Triangle read by rows: T(n,k) is coefficient of k-th power in expansion of ((x^(n+1)-1)/(x-1))^n.

%I A181567 #19 Apr 06 2017 07:52:56
%S A181567 1,1,1,1,2,3,2,1,1,3,6,10,12,12,10,6,3,1,1,4,10,20,35,52,68,80,85,80,
%T A181567 68,52,35,20,10,4,1,1,5,15,35,70,126,205,305,420,540,651,735,780,780,
%U A181567 735,651,540,420,305,205,126,70,35,15,5,1,1,6,21,56,126,252,462,786,1251
%N A181567 Triangle read by rows: T(n,k) is coefficient of k-th power in expansion of ((x^(n+1)-1)/(x-1))^n.
%C A181567 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).
%C A181567 T(n,k) gives the number of divisors of A181555(n) with k prime factors counted with multiplicity. See also A001222, A071207, A146291, A146292.
%C A181567 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
%H A181567 Alois P. Heinz, <a href="/A181567/b181567.txt">Rows n = 0..32, flattened</a>
%H A181567 Anonymous, <a href="http://xrjunque.nom.es/precis/polycalc.aspx">Polynomial calculator</a>
%H A181567 Thomas Wieder, <a href="/A181567/a181567.csv.txt">A181567 as Excel table</a>
%H A181567 G. Xiao, WIMS server, <a href="http://wims.unice.fr/~wims/en_tool~algebra~factor.en.html"> Factoris</a> (both expands and factors polynomials)
%e A181567 Rows begin:
%e A181567 1;
%e A181567 1,1;
%e A181567 1,2,3,2,1;
%e A181567 1,3,6,10,12,12,10,6,3,1;...
%e A181567 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}.
%p A181567 b:= proc(n, k, i) option remember; `if`(k=0, 1,
%p A181567      `if`(i<1, 0, add(b(n, k-j, i-1), j=0..n)))
%p A181567     end:
%p A181567 T:= (n, k)-> b(n, k, n):
%p A181567 seq(seq(T(n, k), k=0..n^2), n=0..8); # _Alois P. Heinz_, Jul 04 2016
%t A181567 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 *)
%Y A181567 A163181 gives row n of n-nomial array. See also A000012, A007318, A027907, A008287, A035343, A063260, A063265, A171890.
%K A181567 easy,nonn,tabf
%O A181567 0,5
%A A181567 _Matthew Vandermast_, Oct 31 2010