A055858 Coefficient triangle for certain polynomials.

1, 1, 2, 4, 9, 6, 27, 64, 48, 36, 256, 625, 500, 400, 320, 3125, 7776, 6480, 5400, 4500, 3750, 46656, 117649, 100842, 86436, 74088, 63504, 54432, 823543, 2097152, 1835008, 1605632, 1404928, 1229312, 1075648, 941192, 16777216, 43046721

Offset: 0

Author

Wolfdieter Lang, Jun 20 2000

Keywords

• easy
• nonn
• tabl

Comments

The coefficients of the partner polynomials are found in triangle A055864.

Examples

			{1}; {1,2}; {4,9,6}; {27,64,48,36}; ...
Fourth row polynomial (n=3): p(3,x) = 27 + 64*x + 48*x^2 + 36*x^3.
		

Crossrefs

Column sequences are A000312(n), n >= 1, A055860 (A000169), A055861 (A053506), A055862-3 for m=0..4, row sums: A045531(n+1)= |A039621(n+1, 2)|, n >= 0.

Programs

Mathematica

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

Formula

a(n, m)=0 if n < m; a(0, 0)=1, a(n, 0) = n^n, n >= 1, a(n, m) = n^(m-1)*(n+1)^(n-m+1), n >= m >= 1;

E.g.f. for column m: A(m, x); A(0, x) = 1/(1+W(-x)); A(1, x) = -1 - (d/dx)W(-x) = -1-W(-x)/((1+W(-x))*x); A(2, x) = A(1, x)-int(A(1, x), x)/x-(1/x+x); recursion: A(m, x) = A(m-1, x)-int(A(m-1, x), x)/x-((m-1)^(m-1))*(x^(m-1))/(m-1)!, m >= 3; W(x) principal branch of Lambert's function.