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

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, 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, 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

Offset: 0

Author

Sela Fried, Sep 19 2021

Keywords

• nonn
• tabf

Comments

The coefficients are ordered lexicographically and by decreasing degree.

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

The sum of each row is n!.

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

The average of each row is n!/C_n.

Examples

			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.
The first six rows of the triangle are:
  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, 2, 1, 1, 4, 9, 6, 3, 6, ...
  ...
		

Links

• Sela Fried, On the coefficients of the distinct monomials in the expansion of x1(x1+x2)...(x1+x2+...+xn), arXiv:2111.07331 [math.CO], 2021.
• R. P. Stanley, Catalan Addendum, 2008.

Crossrefs

Cf. A000108, A000142, A144186, A144187.

Programs

Mathematica

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