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URL: https://oeis.org/A055864

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A055864
Coefficient triangle for certain polynomials.
6
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
1,2
COMMENTS
The coefficients of the partner polynomials are found in triangle A055858.
FORMULA
a(n, m)=0 if n<m; a(n, m)= n^(m-1)*(n+1)^(n-m), 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.
EXAMPLE
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;
...
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 *)
CROSSREFS
Column sequences are: A000272(n+1), n >= 1, A055865, A055070, A055867, A055868 for m=1..5.
Main diagonal gives A000169.
Sequence in context: A338280 A304989 A374756 * A209600 A072045 A189731
KEYWORD
easy,nonn,tabl
AUTHOR
Wolfdieter Lang, Jun 20 2000
STATUS
approved