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

Author

Clark Kimberling

Keywords

• nonn

Comments

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)

Links

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

Programs

Mathematica

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

PARI

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