A067578 -id:A067578 - cp's OEIS frontend

A379149 Specialization of the Elementary Symmetric Functions e(n) at x_i -> Euler phi(i).

1, 1, 1, 1, 2, 1, 1, 4, 5, 2, 1, 6, 13, 12, 4, 1, 10, 37, 64, 52, 16, 1, 12, 57, 138, 180, 120, 32, 1, 18, 129, 480, 1008, 1200, 752, 192, 1, 22, 201, 996, 2928, 5232, 5552, 3200, 768, 1, 28, 333, 2202, 8904, 22800, 36944, 36512, 19968, 4608, 1, 32, 445, 3534, 17712, 58416, 128144, 184288, 166016, 84480, 18432

Offset: 0

Wouter Meeussen, Dec 16 2024

Triangular table with alternating signed sum equal to 0 for n>0,
  1
  1,-1
  1,-2,1
  1,-4,5,-2
  1,-6,13,-12,4
  ..
and with alternating signed weighted sum (first moment) also equal to 0 for n>1,
  0
  0,-1
  0,-2,2
  0,-4,10,-6
  0,-6,26,-36,16
  ..
also when shifting the weights to start at 1,
  1
  1,-2
  1,-4,3
  1,-8,15,-8
  1,-12,39,-48,20

			Triangle begins:
  1;
  1,  1;
  1,  2,  1;
  1,  4,  5,   2;
  1,  6, 13,  12,   4;
  1, 10, 37,  64,  52,  16;
  1, 12, 57, 138, 180, 120, 32;
  ...

Alois P. Heinz, Rows n = 0..140, flattened
Mathematics Stack Exchange, Specializations of elementary symmetric polynomials

Columns k=0-1 give: A000012, A002088.
Main diagonal gives A001088.
T(n,n-1) gives A067578.
Cf. A000010.

b:= proc(n) option remember; `if`(n=0, 1,
       b(n-1)*(1+x*numtheory[phi](n)))
    end:
T:= (n, k)-> coeff(b(n), x, k):
seq(seq(T(n, k), k=0..n), n=0..10);  # Alois P. Heinz, Dec 16 2024

Table[CoefficientList[Expand@Product[z EulerPhi[k]+1,{k,0,n}],z,n+1],{n,0,10}]

row(n) = Vecrev(prod(k=1, n, 1 + 'x * eulerphi(k))) \\ Andrew Howroyd, Dec 16 2024

T(n,k) = [x^k] Product_{j=1..n} (1 + x*phi(j)). - Andrew Howroyd, Dec 16 2024

cp's OEIS Frontend

A379149 Specialization of the Elementary Symmetric Functions e(n) at x_i -> Euler phi(i).

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