0,5

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.

Sequence in context: A131427 A113044 A051883 this_sequence A136572 A053203 A094315

Adjacent sequences: A132789 A132790 A132791 this_sequence A132793 A132794 A132795

easy,nonn

Tom Copeland (tcjpn(AT)msn.com), Nov 17 2007, Nov 27 2007, Nov 29 2007