A078341 Triangle read by rows: T(n,k) = n*T(n-1,k-1) + k*T(n-1,k) starting with T(0,0)=1.

1, 0, 1, 0, 1, 2, 0, 1, 7, 6, 0, 1, 18, 46, 24, 0, 1, 41, 228, 326, 120, 0, 1, 88, 930, 2672, 2556, 720, 0, 1, 183, 3406, 17198, 31484, 22212, 5040, 0, 1, 374, 11682, 96040, 295004, 385144, 212976, 40320, 0, 1, 757, 38412, 489298, 2339380, 4965900

Offset: 1

Author

F. Chapoton, Nov 22 2002

Keywords

• easy
• nonn
• tabl

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.

Examples

			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; ...
		

Crossrefs

Cf. A007549, A000142.

Columns include A000007, A057427, A095151, A103768. Diagonals include A000142, A067318. Row sums are A007549.

Programs

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 *)

Formula

P[1]=1; P[n+1] = x*(d/dx)P[n] + x*n*P[n].

Extensions

Additional comments from Henry Bottomley, Feb 15 2005