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

0,5

Comments

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

Links

Alois P. Heinz, Rows n = 0..140, flattened

Mathematics Stack Exchange, Specializations of elementary symmetric polynomials

Formula

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

Example

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;
  ...

Maple

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

Mathematica

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

Prog

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

Crossrefs

Columns k=0-1 give: A000012, A002088.

Main diagonal gives A001088.

T(n,n-1) gives A067578.

Cf. A000010.

Sequence in context: A263284 A332404 A308905 * A158471 A158472 A198895

Adjacent sequences: A379146 A379147 A379148 * A379150 A379151 A379152

Keyword

nonn,tabl

Author

Wouter Meeussen, Dec 16 2024

Status

approved