A055864 Coefficient triangle for certain polynomials.

1, 3, 2, 16, 12, 9, 125, 100, 80, 64, 1296, 1080, 900, 750, 625, 16807, 14406, 12348, 10584, 9072, 7776, 262144, 229376, 200704, 175616, 153664, 134456, 117649, 4782969, 4251528, 3779136, 3359232, 2985984, 2654208, 2359296, 2097152

Offset: 1

Author

Wolfdieter Lang, Jun 20 2000

Keywords

• easy
• nonn
• tabl

Comments

The coefficients of the partner polynomials are found in triangle A055858.

Examples

			Fourth row polynomial (n=4): p(4,x) = 125+100*x+80*x^2+64*x^3.
Triangle begins:
        1;
        3,       2;
       16,      12,       9;
      125,     100,      80,      64;
     1296,    1080,     900,     750,     625;
    16807,   14406,   12348,   10584,    9072,    7776;
   262144,  229376,  200704,  175616,  153664,  134456,  117649;
  4782969, 4251528, 3779136, 3359232, 2985984, 2654208, 2359296, 2097152;
  ...
		

Crossrefs

Column sequences are: A000272(n+1), n >= 1, A055865, A055070, A055867, A055868 for m=1..5.

Main diagonal gives A000169.

Programs

Mathematica

a[n_, m_] /; n < m = 0; a[n_, m_] := n^(m-1)*(n+1)^(n-m); Table[a[n, m], {n, 1, 8}, {m, 1, n}] // Flatten (* Jean-François Alcover, Jun 20 2013 *)

Formula

a(n, m)=0 if n= m >= 1;

E.g.f. for column m: A(m, x); A(1, x)=-(W(-x)/x+1); recursion: A(m, x) = A(m-1, x)-int(A(m-1, x), x)/x-(((m-1)^(m-1))/m)* (x^(m-1))/(m-1)!, m >= 2; W(x) principal branch of Lambert's function.