A025133 -id:A025133 - cp's OEIS frontend

A025135 (n-1)st elementary symmetric function of binomial(n,0), binomial(n,1), ..., binomial(n,n).

1, 4, 22, 238, 5825, 345600, 51583084, 19765932032, 19661794008192, 51082239411000000, 347836712523276735000, 6221718604078720792473600, 292819054882445795002015111824, 36313083181879002042916296055971840, 11881691691176915544450299522846484375000

Offset: 1

Clark Kimberling

nonn

From R. J. Mathar, Oct 01 2016: (Start)
The k-th elementary symmetric functions of the terms binomial(n,j), j=0..n, form a triangle T(n,k), 0 <= k <= n, n >= 0:
  1
  1   2
  1   4    5
  1   8   22     24
  1  16   93    238     256
  1  32  386   2180    5825     6500
  1  64 1586  19184  117561   345600   407700
  1 128 6476 164864 2229206 15585920 51583084 64538880
  ...
This here is the first subdiagonal. The diagonal is A025134. The 2nd column is A000079, the 2nd A000346, the 3rd A025131, the 4th A025133. (End)

Vincenzo Librandi, Table of n, a(n) for n = 1..70

a[n_] := SymmetricPolynomial[n-1, Table[Binomial[n, k], {k, 0, n}]]; a /@ Range[18] (* Jean-François Alcover, Jul 12 2011 *)

ESym(u)={my(v=vector(#u+1)); v[1]=1; for(i=1, #u, my(t=u[i]); forstep(j=i, 1,-1, v[j+1]+=v[j]*t)); v}
a(n)={ESym(binomial(n))[n]} \\ Andrew Howroyd, Dec 19 2018

A355635 Triangle read by rows. Row n gives the coefficients of Product_{k=0..n-1} (x - binomial(n-1,k)) expanded in decreasing powers of x, with row 0 = {1}.

Original entry on oeis.org

1, 1, -1, 1, -2, 1, 1, -4, 5, -2, 1, -8, 22, -24, 9, 1, -16, 93, -238, 256, -96, 1, -32, 386, -2180, 5825, -6500, 2500, 1, -64, 1586, -19184, 117561, -345600, 407700, -162000, 1, -128, 6476, -164864, 2229206, -15585920, 51583084, -64538880, 26471025

Offset: 0

Thomas Scheuerle, Jul 11 2022

Without signs the triangle of elementary symmetric functions of the terms binomial(n,j), j=0..n.

			The triangle begins:
  1;
  1,  -1;
  1,  -2,   1;
  1,  -4,   5,    -2;
  1,  -8,  22,   -24,    9;
  1, -16,  93,  -238,  256,   -96;
  1, -32, 386, -2180, 5825, -6500, 2500;
  ...
Row 4: x^4 - 8*x^3 + 22*x^2 - 24*x + 9 = (x-1)*(x-4)*(x-6)*(x-4)*(x-1).

Cf. A000079, A000346, A025131, A025133, A025134, A025135.

Cf. A001142 (right diagonal unsigned).

T(n, k) = polcoeff(prod(m=0, n, (x-binomial(n-1, m))), n-k+1);

T(n, 0) = 1.
T(n, 1) = -2^(n-1), for n > 0.
T(n, 2) = A000346(n-2), for n > 1.
T(n, 3) = -A025131(n-1), for n > 1.
T(n, 4) = A025133(n-1), for n > 1.
T(n, n) = (-1)^n*A001142(n-1), for n > 0.
T(n+1, n) = (-1)^n*A025134(n).
T(n+2, n) = (-1)^n*A025135(n).

cp's OEIS Frontend

A025135 (n-1)st elementary symmetric function of binomial(n,0), binomial(n,1), ..., binomial(n,n).

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