OFFSET
1,6
COMMENTS
Triangle of coefficients of polynomials P[n]. Let F(t) satisfy dF/dt = exp(x*(exp(F)-1)) and F(0)=0. Then F(t) = Sum_{n>=0} P[n]/n! t^n, where P[n] is a polynomial in x of degree n-1. The constant term of the polynomial is zero for n >= 2. The coefficient of x is 1 for n >= 2. The coefficient of x^n in P[n+1] is n!. The value at 1 is given by sequence A007549.
FORMULA
P[1]=1; P[n+1] = x*(d/dx)P[n] + x*n*P[n].
EXAMPLE
P[1]=1, P[2]=x, P[3]=x+2*x^2, P[4]=x+7*x^2+6*x^3, P[5]=x+18*x^2+46*x^3+24*x^4, P[6]=x+41*x^2+228*x^3+326*x^4+120*x^5.
Rows start 1; 0,1; 0,1,2; 0,1,7,6; 0,1,18,46,24; 0,1,41,228,326,120; ...
MAPLE
P[1] := 1; for n from 1 to 10 do P[n+1] := expand(x*diff(P[n], x)+x*n*P[n]) od;
MATHEMATICA
p[1][x_] = 1; p[n_][x_] := x*p[n-1]'[x] + x*(n-1)*p[n-1][x]; Table[ CoefficientList[ p[n][x], x], {n, 1, 10}] // Flatten (* Jean-François Alcover, Jan 29 2013 *)
CROSSREFS
KEYWORD
AUTHOR
F. Chapoton, Nov 22 2002
EXTENSIONS
Additional comments from Henry Bottomley, Feb 15 2005
STATUS
approved