OFFSET
0,9
COMMENTS
Matrix T begins
0;
0,0;
0,1,0;
0,0,2,0;
0,0,0,3,0;
0,0,0,0,4,0;
Let M(t) = exp(t*T) = limit [1 + t*T/n]^n as n tends to infinity.
Then M(1) = the lower triangular padded Pascal matrix A097805, with inverse M(-1).
Given a polynomial sequence p_n(x) with p_0(x)=1 and the lowering and raising operators L and R defined by L P_n(x) = n * P_(n-1)(x) and R P_n(x) = P_(n+1)(x), the matrix T represents the action of R^2*L in the p_n(x) basis. For p_n(x) = x^n, L = D = d/dx and R = x. For p_n(x) = x^n/n!, L = DxD and R = D^(-1).
See A132440 for an analog and more general discussion.
LINKS
P. Blasiak and P. Flajolet, Combinatorial models of creation-annihilation
T. Copeland, Mathemagical Forests
T. Copeland, Addendum to Mathemagical Forests
G. Dattoli, B. Germano, M. Martinelli, and P. Ricci, Touchard like polynomials and generalized Stirling polynomials
FORMULA
The matrix operation b = T*a can be characterized in several ways in terms of the coefficients a(n) and b(n), their o.g.f.s A(x) and B(x), or e.g.f.s EA(x) and EB(x):
1) b(0) = 0, b(1) = 0, b(n) = (n-1) * a(n-1),
2) B(x) = x^2D A(x)= x (xDx)(1/x)A(x) = x^2 * Lag(1,-:xD:) A(x)/x , or
3) EB(x) = D^(-1)xD EA(x),
where D is the derivative w.r.t. x, (D^(-1)x^j/j!) = x^(j+1)/(j+1)!, (:xD:)^j = x^j*D^j, and Lag(n,x) are the Laguerre polynomials A021009.
So the exponentiated operator can be characterized as
4) exp(t*T) A(x) = exp(t*x^2D) A(x) = x exp(t*xDx)(1/x)A(x)
= x [sum(n=0,1,...) (t*x)^n * Lag(n,-:xD:)] A(x)/x
= x [exp{[t*u/(1-t*u)]*:xD:} / (1-t*u) ] A(x)/x (eval. at u=x)
= A[x/(1-t*x)], a special Moebius or linear fractional trf.,
5) exp(t*T) EA(x) = D^(-1) exp(t*x)D EA(x), a shifted Euler trf.
for an e.g.f., or
6) [exp(t*T) * a]_n = [M(t) * a]_n
= [sum(k=0,...,n-1) binomial(n-1,k)* t^(n-1-k) * a(k+1)] with [M(t) * a]_0 = a_0
MATHEMATICA
Table[PadLeft[{n-1, 0}, n+1], {n, 0, 12}] // Flatten (* Jean-François Alcover, Apr 30 2014 *)
CROSSREFS
KEYWORD
AUTHOR
Tom Copeland, Oct 24 2012
STATUS
approved