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%I #11 Jan 08 2018 21:02:19
%S 1,1,2,4,9,6,27,64,48,36,256,625,500,400,320,3125,7776,6480,5400,4500,
%T 3750,46656,117649,100842,86436,74088,63504,54432,823543,2097152,
%U 1835008,1605632,1404928,1229312,1075648,941192,16777216,43046721
%N Coefficient triangle for certain polynomials.
%C The coefficients of the partner polynomials are found in triangle A055864.
%F 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;
%F 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.
%e {1}; {1,2}; {4,9,6}; {27,64,48,36}; ...
%e Fourth row polynomial (n=3): p(3,x) = 27 + 64*x + 48*x^2 + 36*x^3.
%t 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 *)
%Y 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.
%K easy,nonn,tabl
%O 0,3
%A _Wolfdieter Lang_, Jun 20 2000