h(k, x) = (-I/sqrt(2))^k * H(k, I*x/sqrt(2)), H(n, x) the Hermite polynomials (
A060821,
A059343).
T(n,k) = n!/(2^((n-k)/2)*((n-k)/2)!k!) if n-k >= 0 is even; 0 otherwise. -
Emeric Deutsch, Oct 14 2006
G.f.: 1/(1-x*y-x^2/(1-x*y-2*x^2/(1-x*y-3*x^2/(1-x*y-4*x^2/(1-... (continued fraction). -
Paul Barry, Apr 10 2009
Recurrence: T(0,0)=1, T(0,k)=0 for k>0 and for n >= 1 T(n,k) = T(n-1,k-1) + (k+1)*T(n-1,k+1). -
Peter Luschny, Oct 06 2012
The row polynomials P(n,x) = (a. + x)^n, umbrally evaluated with (a.)^n = a_n = aerated
A001147, are an Appell sequence with dP(n,x)/dx = n * P(n-1,x). The umbral compositional inverses (cf.
A001147) of these polynomials are given by the same polynomials signed,
A066325. -
Tom Copeland, Nov 15 2014
The odd rows are (2x^2)^n x n! L(n,-1/(2x^2),1/2), and the even, (2x^2)^n n! L(n,-1/(2x^2),-1/2) in sequence with n= 0,1,2,... and L(n,x,a) = Sum_{k=0..n} binomial(n+a,k+a) (-x)^k/k!, the associated Laguerre polynomial of order a. The odd rows are related to
A130757, and the even to
A176230 and
A176231. Other versions of this entry are
A122848,
A049403,
A096713 and
A104556, and reversed
A100861,
A144299,
A111924. With each non-vanishing diagonal divided by its initial element
A001147(n), this array becomes reversed, aerated
A034839.
Create four shift and stretch matrices S1,S2,S3, and S4 with all elements zero except S1(2n,n) = 1 for n >= 1, S2(n,2n) = 1 for n >= 0, S3(2n+1,n) = 1 for n >= 1, and S4(n,2n+1) = 1 for n >= 0. Then this entry's lower triangular matrix is T = Id + S1 * (
A176230-Id) * S2 + S3 * (unsigned
A130757-Id) * S4 with Id the identity matrix. The sandwiched matrices have infinitesimal generators with the nonvanishing subdiagonals
A000384(n>0) and
A014105(n>0).
As an Appell sequence, the lowering and raising operators are L = D and R = x + dlog(exp(D^2/2))/dD = x + D, where D = d/dx, L h(n,x) = n h(n-1,x), and R h(n,x) = h(n+1,x), so R^n 1 = h(n,x). The fundamental moment sequence has the e.g.f. e^(t^2/2) with coefficients a(n) = aerated
A001147, i.e., h(n,x) = (a. + x)^n, as noted above. The raising operator R as a matrix acting on o.g.f.s (formal power series) is the transpose of the production matrix P below, i.e., (1,x,x^2,...)(P^T)^n (1,0,0,...)^T = h(n,x).
For characterization as a Riordan array and associations to combinatorial structures, see the Barry link and the Yang and Qiao reference. For relations to projective modules, see the Sazdanovic link.
(End)
From the Appell formalism, e^(D^2/2) x^n = h_n(x), the n-th row polynomial listed below, and e^(-D^2/2) x^n = u_n(x), the n-th row polynomial of
A066325. Then R = e^(D^2/2) * x * e^(-D^2/2) is another representation of the raising operator, implied by the umbral compositional inverse relation h_n(u.(x)) = x^n. -
Tom Copeland, Oct 02 2016
h_n(x) = p_n(x-1), where p_n(x) are the polynomials of
A111062, related to the telephone numbers
A000085. -
Tom Copeland, Jun 26 2018
In the power basis x^n, the matrix infinitesimal generator M =
A132440^2/2, when acting on a row vector for an o.g.f., is the matrix representation for the differential operator D^2/2.
e^{M} gives the coefficients of the Hermite polynomials of this entry.
The only nonvanishing subdiagonal of M, the second subdiagonal (1,3,6,10,...), gives, aside from the initial 0, the triangular numbers
A000217, the number of edges of the n-dimensional simplices with (n+1) vertices. The perfect matchings of these simplices are the aerated odd double factorials
A001147 noted above, the moments for the Hermite polynomials.
The polynomials are also generated from
A036040 with x[1] = x, x[2] = 1, and the other indeterminates equal to zero. (End)
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