A055867 -id:A055867 - cp's OEIS frontend

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

Wolfdieter Lang, Jun 20 2000

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

			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;
  ...

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

Main diagonal gives A000169.

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 *)

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.

A055868 Fifth column of triangle A055864.

Original entry on oeis.org

0, 0, 0, 0, 625, 9072, 153664, 2985984, 65610000, 1610510000, 43717791744, 1301153248512, 42150011228416, 1476840093750000, 55662776156160000, 2246042996417036288, 96619713808095645696, 4414553991933897025584

Offset: 1

Wolfdieter Lang Jun 20 2000

a(n)=A055864(n, 5). Cf. A000272, A055865, A055070, A055867, A055858.

a(i)=0 for i=1..4; a(n)= n^4*(n+1)^(n-5), n >= 5.

E.g.f.: (270*W(-x)^2-740*W(-x)^3+345*W(-x)^4-36*W(-x)^5-270*x^2-1280*x^3-3645*x^4-9216*x^5)/(4320*x), W(x) principal branch of Lambert's function.

cp's OEIS Frontend

A055864 Coefficient triangle for certain polynomials.

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