A132792 The infinitesimal Lah matrix: generator of unsigned A111596.

0, 0, 0, 0, 2, 0, 0, 0, 6, 0, 0, 0, 0, 12, 0, 0, 0, 0, 0, 20, 0, 0, 0, 0, 0, 0, 30, 0, 0, 0, 0, 0, 0, 0, 42, 0, 0, 0, 0, 0, 0, 0, 0, 56, 0, 0, 0, 0, 0, 0, 0, 0, 0, 72, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 90, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 110, 0

Offset: 0

Author

Tom Copeland, Nov 17 2007, Nov 27 2007, Nov 29 2007

Keywords

• easy
• nonn
• tabl

Comments

The matrix T begins

0;

0, 0;

0, 2, 0;

0, 0, 6, 0;

0, 0, 0, 12, 0;

Along the nonvanishing diagonal the n-th term is (n+1)*(n).

Let LM(t) = exp(t*T) = limit [1 + t*T/n]^n as n tends to infinity.

Lah matrix = [ bin(n,k)*(n-1)!/(k-1)! ] = LM(1) = exp(T) = unsigned A111596. Truncating the series gives the n X n principal submatrices. In fact, the principal submatrices of T are nilpotent with [Tsub_n]^n = 0 for n=0,1,2,....

Inverse Lah matrix = LM(-1) = exp(-T)

Umbrally LM[b(.)] = exp(b(.)*T) = [ bin(n,k)*(n-1)!/(k-1)! * b(n-k) ]

A(j) = T^j / j! equals the matrix [ bin(n,k)*(n-1)!/(k-1)! * delta(n-k-j)] where delta(n) = 1 if n=0 and vanishes otherwise (Kronecker delta); i.e. A(j) is a matrix with all the terms 0 except for the j-th lower (or main for j=0) diagonal which equals that of the Lah matrix. Hence the A(j)'s form a linearly independent basis for all matrices of the form [ bin(n,k)*(n-1)!/(k-1)! * d(n-k) ].

For sequences with b(0) = 1, umbrally,

LM[b(.)] = exp(b(.)*T) = [ bin(n,k)*(n-1)!/(k-1)! * b(n-k) ] .

[LM[b(.)]]^(-1) = exp(c(.)*T) = [ bin(n,k)*(n-1)!/(k-1)! * c(n-k) ] where c = LPT(b) with LPT the list partition transform of A133314. Or,

[LM[b(.)]]^(-1) = exp[LPT(b(.))*T] = LPT[LM(b(.))] = LM[LPT(b(.))] = LM[c(.)] .

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(n) = n*(n-1) * a(n-1),

2) B(x) = [ x^2 * D^2 * x ] A(x)

3) B(x) = [ x^2 * 2 * Lag(2,-:xD:,0) x^(-1) ] A(x)

4) EB(x) = [ D^(-1) * x * D^2 * x ] EA(x)

where D is the derivative w.r.t. x, (:xD:)^j = x^j * D^j and Lag(n,x,m) is the associated Laguerre polynomial of order m.

The exponentiated operator can be characterized (with loose notation) as

5) exp(t*T) * a = LM(t) * a = [sum(k=0,...,n) bin(n-1,k-1) * (n! / k!) t^(n-k) * a(k) ] = [ t^n * n! * Lag(n,-a(.)/t,-1) ], a vector array. Note binomial(n-1,k-1) is 1 for n=k=0 and vanishes for n>0 and k=0 .

With t=1 and a(k) = (-x)^k, then LM(1) * a = [ n! * Laguerre(n,x,-1) ], a vector array with index n .

6) exp(t*T) EA(x) = EB(x) = EA[ x / (1-x*t) ]

From the inverse operator (change t to -t), inverting amounts to substituting x/(1+x*t) for x in EB(x) in formula 6.

Compare analogous results in A132710.

T is also a shifted version of the infinitesimal Pascal matrix squared, i.e., T = (A132440^2) * A129185 . The non-vanishing diagonal of T is A002378.

Links

• M. Janjic, Some classes of numbers and derivatives, JIS 12 (2009) 09.8.3

Programs

Mathematica

Table[PadLeft[{n*(n-1), 0}, n+1], {n, 0, 11}] // Flatten (* Jean-François Alcover, Apr 30 2014 *)

Formula

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^2*R

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). - Tom Copeland, Oct 25 2012