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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, 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.
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
KEYWORD
nonn,tabf
AUTHOR
Sela Fried, Sep 19 2021
STATUS
approved