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

0,6

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.

Links

Table of n, a(n) for n=0..104.

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.

Example

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, ...
  ...

Mathematica

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

Crossrefs

Cf. A000108, A000142, A144186, A144187.

Sequence in context: A196228 A133364 A063420 * A254631 A029385 A185155

Adjacent sequences: A347914 A347915 A347916 * A347918 A347919 A347920

Keyword

nonn,tabf

Author

Sela Fried, Sep 19 2021

Status

approved