%I A132792
%S A132792 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,
%T A132792 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,
%U A132792 0,0,0,0,0,0,0,0,0,110,0
%N A132792 The infinitesimal Lah matrix: generator of unsigned A111596.
%C A132792 The matrix T begins
%C A132792 0;
%C A132792 0, 0;
%C A132792 0, 2, 0;
%C A132792 0, 0, 6, 0;
%C A132792 0, 0, 0, 12, 0;
%C A132792 Along the nonvanishing diagonal the n-th term is (n+1)*(n).
%C A132792 Let LM(t) = exp(t*T) = limit [1 + t*T/n]^n as n tends to infinity.
%C A132792 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,....
%C A132792 Inverse Lah matrix = LM(-1) = exp(-T)
%C A132792 Umbrally LM[b(.)] = exp(b(.)*T) = [ bin(n,k)*(n-1)!/(k-1)! * b(n-k) ]
%C A132792 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) ].
%C A132792 For sequences with b(0) = 1, umbrally,
%C A132792 LM[b(.)] = exp(b(.)*T) = [ bin(n,k)*(n-1)!/(k-1)! * b(n-k) ] .
%C A132792 [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,
%C A132792 [LM[b(.)]]^(-1) = exp[LPT(b(.))*T] = LPT[LM(b(.))] = LM[LPT(b(.))] = 
               LM[c(.)] .
%C A132792 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).
%C A132792 1) b(0) = 0, b(n) = n*(n-1) * a(n-1),
%C A132792 2) B(x) = [ x^2 * D^2 * x ] A(x)
%C A132792 3) B(x) = [ x^2 * 2 * Lag(2,-:xD:,0) x^(-1) ] A(x)
%C A132792 4) EB(x) = [ D^(-1) * x * D^2 * x ] EA(x)
%C A132792 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.
%C A132792 The exponentiated operator can be characterized (with loose notation) 
               as
%C A132792 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 
               .
%C A132792 With t=1 and a(k) = (-x)^k, then LM(1) * a = [ n! * Laguerre(n,x,-1) 
               ], a vector array with index n .
%C A132792 6) exp(t*T) EA(x) = EB(x) = EA[ x / (1-x*t) ]
%C A132792 From the inverse operator (change t to -t), inverting amounts to substituting 
               x/(1+x*t) for x in EB(x) in formula 6.
%C A132792 Compare analogous results in A132710.
%C A132792 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.
%Y A132792 Sequence in context: A113044 A082399 A051883 this_sequence A136572 A053203 
               A158360
%Y A132792 Adjacent sequences: A132789 A132790 A132791 this_sequence A132793 A132794 
               A132795
%K A132792 easy,nonn
%O A132792 0,5
%A A132792 Tom Copeland (tcjpn(AT)msn.com), Nov 17 2007, Nov 27 2007, Nov 29 2007