A055864 - OEIS

%I #11 Jun 26 2024 15:20:30

%S 1,3,2,16,12,9,125,100,80,64,1296,1080,900,750,625,16807,14406,12348,

%T 10584,9072,7776,262144,229376,200704,175616,153664,134456,117649,

%U 4782969,4251528,3779136,3359232,2985984,2654208,2359296,2097152

%N Coefficient triangle for certain polynomials.

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

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

%F 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.

%e Fourth row polynomial (n=4): p(4,x) = 125+100*x+80*x^2+64*x^3.

%e Triangle begins:

%e         1;

%e         3,       2;

%e        16,      12,       9;

%e       125,     100,      80,      64;

%e      1296,    1080,     900,     750,     625;

%e     16807,   14406,   12348,   10584,    9072,    7776;

%e    262144,  229376,  200704,  175616,  153664,  134456,  117649;

%e   4782969, 4251528, 3779136, 3359232, 2985984, 2654208, 2359296, 2097152;

%e   ...

%t 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 *)

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

%Y Main diagonal gives A000169.

%K easy,nonn,tabl

%O 1,2

%A _Wolfdieter Lang_, Jun 20 2000