A347917 - OEIS

%I #31 Nov 17 2021 12:38:33

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

%C The coefficients are ordered lexicographically and by decreasing degree.

%C Each row of the triangle consists of C_n numbers where C_n is the n-th Catalan number.

%C The sum of each row is n!.

%C In the triangle, the (n+1)-th row contains (at least) two copies of the n-th row.

%C The average of each row is n!/C_n.

%H 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 R. P. Stanley, <a href="http://www-math.mit.edu/~rstan/ec#catadd">Catalan Addendum</a>, 2008.

%e 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 The first six rows of the triangle are:

%e   1

%e   1

%e   1, 1

%e   1, 2, 1, 1, 1

%e   1, 3, 2, 1, 3, 4, 2, 1, 1, 1, 2, 1, 1, 1

%e   1, 4, 3, 2, 1, 6, 9, 6, 3, 3, 4, 2, 1, 1, 4, 9, 6, 3, 6, ...

%e   ...

%t Join@@Table[Values@CoefficientRules[Times@@Array[Total@Array[x,#]&,n]],{n,6}] (* _Giorgos Kalogeropoulos_, Nov 16 2021 *)

%Y Cf. A000108, A000142, A144186, A144187.

%K nonn,tabf

%O 0,6

%A _Sela Fried_, Sep 19 2021