0,2

Analogous to the infinitesimal generators of A132681 and A132792.

The matrix T begins

0;

2,0;

0, 6, 0;

0, 0, 12 0;

0, 0, 0, 20, 0;

Along the nonvanishing diagonal the n-th term is (n+2)*(n+1).

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

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

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

Umbrally shifted Lah[b(.)] = exp(b(.)*T) = [ binomial(n+1,k+1)*(n)!/(k)! * b(n-k) ]

A(j) = T^j / j! equals the matrix [binomial(n+1,k+1)*(n)!/(k)! * 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 [binomial(n+1,k+1) * (n)! / (k)! * d(n-k)].

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

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

[LM[b(.)]]^(-1) = exp(c(.)*T) = [ bin(n+1,k+1)*(n)!/(k)! * c(n-k) ] where c = LPT(b) with LPT the list partition transform of 133314. 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+1)*(n) * a(n-1),

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

3) B(x) = x * 2 *Lag(2,-:xD:,0) A(x)

4) EB(x) = D * x^2 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.

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) ] / (1-x*t)^2

Sequence in context: A095221 A078112 A128711 this_sequence A106512 A094785 A035536

Adjacent sequences: A132707 A132708 A132709 this_sequence A132711 A132712 A132713

easy,nonn

Tom Copeland (tcjpn(AT)msn.com), Nov 15 2007, Nov 16 2007, Nov 27 2007