%I A132014
%S A132014 1,2,1,2,4,1,0,6,6,1,0,0,12,8,1,0,0,0,20,10,1,0,0,0,0,30,12,1
%V A132014 1,-2,1,2,-4,1,0,6,-6,1,0,0,12,-8,1,0,0,0,20,-10,1,0,0,0,0,30,-12,1
%N A132014 T(n,j) for double application of an iterated mixed order Laguerre transform.
%C A132014 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 A132014 1) b(0) = a(0), b(1) = a(n) - 2 a(0), b(n) = a(n) - 2n a(n-1) + n(n-1) 
               a(n-2) for n > 0.
%C A132014 2) b(n) = n! Lag{n,(.)!*Lag[.,a1(.),0],-1}, umbrally, where a1(n) = n! 
               Lag{n,(.)!*Lag[.,a(.),0],-1} .
%C A132014 3) b(n) = n! sum(j=0,...,min(2,n)) (-1)^j * Binom(n,j)*a(n-j)/(n-j)!
%C A132014 4) b(n) = (-1)^n n! Lag(n,a(.),2-n)
%C A132014 5) B(x) = (1-xDx))^2 A(x)
%C A132014 6) B(x) = sum(j=0,1,2) {(-1)^j * Binom(2,j)*j!*x^j*Lag(j,-:xD:,0)} A(x)
%C A132014 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 A132014 7) EB(x) = (1-x)^2 EA(x)
%C A132014 8) T = S^2 = A132013^2 = A094587^(-2) = A132159^(-1).
%C A132014 c = (1,-2,2,0,0,...) is the sequence associated to T under the list partition 
               transform and associated operations described in A133314. c are also 
               the coefficients in formula 6. Thus T(n,k) = binomial(n,k)*c(n-k).
%C A132014 The reciprocal sequence to c is d = (1!,2!,3!,4!,...), so the inverse 
               of T is TI(n,k) = binomial(n,k)*d(n-k) = A132159.
%C A132014 These formulae are easily generalized for m applications of the basic 
               operator n! Lag[n,(.)!*Lag[.,a(.),0],-1] by replacing 2 by m in formulae 
               3, 4, 5, 6 and 7.
%C A132014 The generalized c are given by the generalized coefficients of 6, i.e.,
%C A132014 c(n) = (-1)^n * Binom(m,n)*n! = (-1)^n * m!/(m-n)! .
%C A132014 The generalized d are given by the array at and below the term SI(m-1,
               m-1) in SI(n,k) = Binom(n,k) * (n-k)!, the inverse of S ; i.e.,
%C A132014 d(n) = SI(m-1+n,m-1) = Binom(m-1+n,m-1) * n! = (m-1+n)!/(m-1)! .
%C A132014 As an aside, this shows that the signed falling factorials and the rising 
               factorials form reciprocal pairs under the list partition transform 
               of A133314.
%C A132014 Row sums of T = [formula 3 with all a(n) = 1] = [binomial transform of 
               c] = [coefficients of B(x) with A(x) = 1/(1-x)] = (1,-1,-1,1,5,11,
               19,...),
%C A132014 with e.g.f. = [EB(x) with EA(x) = exp(x)] = (1-x)^2 * exp(x) = exp(x)*exp(c(.)*x) 
               = exp[(1+c(.))*x].
%C A132014 Alternating row sums of T = [formula 3 with all a(n) = (-1)^n] = [finite 
               differences of c] = [coefficients of B(x) with A(x) = 1/(1+x)] = 
               (1,-3,7,-13,21,-31,...) = (-1)^n A002061(n+1),
%C A132014 with e.g.f. = [EB(x) with EA(x) = exp(-x)] = (1-x)^2 * exp(-x) = exp(- 
               x)*exp(c(.)*x) = exp[-(1-c(.))*x].
%F A132014 T(n,k) = binomial(n,k)*c(n-k)
%Y A132014 Sequence in context: A137408 A007461 A143446 this_sequence A110330 A097864 
               A097866
%Y A132014 Adjacent sequences: A132011 A132012 A132013 this_sequence A132015 A132016 
               A132017
%K A132014 easy,sign
%O A132014 0,2
%A A132014 Tom Copeland (tcjpn(AT)msn.com), Oct 30 2007, Nov 05 2007, Nov 11 2007