

A132014


T(n,j) for double application of an iterated mixed order Laguerre transform: Coefficients of Laguerre polynomial (1)^n*n!*L(n,2n,x)


8



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, 0, 0, 0, 0, 0, 42, 14, 1, 0, 0, 0, 0, 0, 0, 56, 16, 1, 0, 0, 0, 0, 0, 0, 0, 72, 18, 1, 0, 0, 0, 0, 0, 0, 0, 0, 90, 20, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 110, 22, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 132, 24, 1
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OFFSET

0,2


COMMENTS

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) = a(0), b(1) = a(n)  2 a(0), b(n) = a(n)  2n a(n1) + n(n1) a(n2) for n > 0.
2) b(n) = n! Lag{n,(.)!*Lag[.,a1(.),0],1}, umbrally, where a1(n) = n! Lag{n,(.)!*Lag[.,a(.),0],1}.
3) b(n) = n! sum(j=0,...,min(2,n)) (1)^j * Binom(n,j)*a(nj)/(nj)!
4) b(n) = (1)^n n! Lag(n,a(.),2n)
5) B(x) = (1xDx)^2 A(x)
6) B(x) = sum(j=0,1,2) {(1)^j * Binom(2,j)*j!*x^j*Lag(j,:xD:,0)} A(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.
7) EB(x) = (1x)^2 EA(x)
8) T = S^2 = A132013^2 = A094587^(2) = A132159^(1).
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(nk).
The reciprocal sequence to c is d = (1!,2!,3!,4!,...), so the inverse of T is TI(n,k) = binomial(n,k)*d(nk) = A132159.
These formulas are easily generalized for m applications of the basic operator n! Lag[n,(.)!*Lag[.,a(.),0],1] by replacing 2 by m in formulas 3, 4, 5, 6 and 7.
The generalized c are given by the generalized coefficients of 6, i.e.,
c(n) = (1)^n * Binom(m,n)*n! = (1)^n * m!/(mn)!.
The generalized d are given by the array at and below the term SI(m1,m1) in SI(n,k) = Binom(n,k) * (nk)!, the inverse of S; i.e.,
d(n) = SI(m1+n,m1) = Binom(m1+n,m1) * n! = (m1+n)!/(m1)!.
As an aside, this shows that the signed falling factorials and the rising factorials form reciprocal pairs under the list partition transform of A133314.
Row sums of T = [formula 3 with all a(n) = 1] = [binomial transform of c] = [coefficients of B(x) with A(x) = 1/(1x)] = (1,1,1,1,5,11,19,...),
with e.g.f. = [EB(x) with EA(x) = exp(x)] = (1x)^2 * exp(x) = exp(x)*exp(c(.)*x) = exp[(1+c(.))*x].
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),
with e.g.f. = [EB(x) with EA(x) = exp(x)] = (1x)^2 * exp(x) = exp( x)*exp(c(.)*x) = exp[(1c(.))*x].
See A132159 for a relation to the PoissonCharlier polynomials.  Tom Copeland, Jan 15 2016


LINKS

Table of n, a(n) for n=0..90.
M. Janjic, Some classes of numbers and derivatives, JIS 12 (2009) 09.8.3
Wikipedia, Appell sequence


FORMULA

T(n,k) = binomial(n,k)*c(nk)
E.g.f. for row polynomials: exp(x*y)(1x)^2. Implies the row polynomials form an Appell sequence (see Wikipedia).  Tom Copeland, Dec 03 2013
From Tom Copeland, Apr 21 2014: (Start)
Change notation letting L(n,m,x) = Lag(n,x,m).
Row polynomials: (1)^n*n!*L(n,2n,x) = (1)^n*(x)^(n2)*2!*L(2,n2,x) =
(1)^n*(2!/(2n)!)*K(n,2n+1,x) where K is Kummer's confluent hypergeometric function (as a limit of n+s as s tends to zero).
For the row polynomials, the lowering operator = d/dx and the raising operator = x  2/(1D).
T = (I  A132440)^2 = [2*I  exp(A238385I)]^2 = signed exp[2*(A238385I)], where I = identity matrix.
Operationally, (1)^n*n!*L(n,2n,:xD:) = (1)^n*x^(n2)*:Dx:^n*x^(2n) = (1)^n*x^(2)*:xD:^n*x^2 = (1)^n*n!*Binom(xD+2,n) = (1)^n*n!*Binom(2,n)*K(n,2n+1,:xD:) where :AB:^n = A^n*B^n for any two operators. Cf. A235706. (End)


EXAMPLE

First few rows of the triangle are
1;
2, 1;
2, 4, 1;
0, 6, 6, 1;
0, 0, 12, 8, 1;
0, 0, 0, 20, 10, 1;


MATHEMATICA

m = 12; s = Exp[x*y]*(1  x)^2 + O[x]^(m + 2) + O[y]^(m + 2); T[n_, k_] := SeriesCoefficient[s, {x, 0, n}, {y, 0, k}]*n!; T[0, 0] = 1; Table[T[n, k], {n, 0, m}, {k, 0, n}] // Flatten (* JeanFrançois Alcover, Jul 09 2015 *)


CROSSREFS

Cf. A132382, A110330, A132159, A094587, A132013.
Sequence in context: A181302 A143446 A110330 * A097864 A097866 A097865
Adjacent sequences: A132011 A132012 A132013 * A132015 A132016 A132017


KEYWORD

easy,sign,tabl


AUTHOR

Tom Copeland, Oct 30 2007, Nov 05 2007, Nov 11 2007


EXTENSIONS

Title modified by Tom Copeland, Apr 21 2014
Missing term 18 inserted in 10th row by JeanFrançois Alcover, Jul 09 2015


STATUS

approved



