cp's OEIS Frontend

A347917 The coefficients in the expansion x_1(x_1 + x_2)...(x_1 + x_2 + ... + x_n), given row by row.

%I A347917 #31 Nov 17 2021 12:38:33
%S A347917 1,1,1,1,1,2,1,1,1,1,3,2,1,3,4,2,1,1,1,2,1,1,1,1,4,3,2,1,6,9,6,3,3,4,
%T A347917 2,1,1,4,9,6,3,6,8,4,2,2,1,2,1,1,1,1,3,2,1,3,4,2,1,1,1,2,1,1,1,1,5,4,
%U A347917 3,2,1,10,16,12,8,4,6,9,6,3,3,4,2,1,1,10,24,18,12,6,18,27,18,9,9,12,6,3,3,4,9,6,3,6,8
%N A347917 The coefficients in the expansion x_1(x_1 + x_2)...(x_1 + x_2 + ... + x_n), given row by row.
%C A347917 The coefficients are ordered lexicographically and by decreasing degree.
%C A347917 Each row of the triangle consists of C_n numbers where C_n is the n-th Catalan number.
%C A347917 The sum of each row is n!.
%C A347917 In the triangle, the (n+1)-th row contains (at least) two copies of the n-th row.
%C A347917 The average of each row is n!/C_n.
%H A347917 Sela Fried, <a href="https://arxiv.org/abs/2111.07331">On the coefficients of the distinct monomials in the expansion of x1(x1+x2)...(x1+x2+...+xn)</a>, arXiv:2111.07331 [math.CO], 2021.
%H A347917 R. P. Stanley, <a href="http://www-math.mit.edu/~rstan/ec#catadd">Catalan Addendum</a>, 2008.
%e A347917 The fourth row of the triangle is 1,2,1,1,1 since x_1(x_1 + x_2)(x_1 + x_2 + x_3) = x_1^3 + 2x_1^2x_2+x_1x_2^2 + x_1^2x_3+x_1x_2x_3.
%e A347917 The first six rows of the triangle are:
%e A347917   1
%e A347917   1
%e A347917   1, 1
%e A347917   1, 2, 1, 1, 1
%e A347917   1, 3, 2, 1, 3, 4, 2, 1, 1, 1, 2, 1, 1, 1
%e A347917   1, 4, 3, 2, 1, 6, 9, 6, 3, 3, 4, 2, 1, 1, 4, 9, 6, 3, 6, ...
%e A347917   ...
%t A347917 Join@@Table[Values@CoefficientRules[Times@@Array[Total@Array[x,#]&,n]],{n,6}] (* _Giorgos Kalogeropoulos_, Nov 16 2021 *)
%Y A347917 Cf. A000108, A000142, A144186, A144187.
%K A347917 nonn,tabf
%O A347917 0,6
%A A347917 _Sela Fried_, Sep 19 2021