cp's OEIS Frontend

The Copeland link gives the associations of this entry with the operator calculus of Appell Sheffer polynomials, the combinatorics of simple set partitions encoded in the Faa di Bruno formula for composition of analytic functions (formal Taylor series), the Pascal matrix, and the geometry of the n-dimensional simplices (hypertriangles, or hypertetrahedra). These, in turn, are related to simple instances of the application of the exponential formula / principle / schema giving the number of not-necessarily-connected objects composed from an ensemble of connected objects. - Tom Copeland, Jun 09 2021

From Robert Israel, Apr 27 2017: (Start)

(-1)^n*a(n) is (the number of even involutions) - (the number of odd involutions) in the symmetric group S_n.

a(n) == (-1)^n (mod A069834(n-1)) for n >= 3.

a(n) is divisible by n-2 and by A200675(n+2). (End)

Also number of involutions on n labeled elements with k fixed points times (-1)^(number of 2-cycles).

Also called normalized Hermite polynomials.

He_n(x) := H_n(x/sqrt(2)) / sqrt(2)^n, with the coefficients of H_n(x) given in A060821. See the Maple program. - Wolfdieter Lang, Jan 13 2020

From R. J. Mathar, Jun 09 2008: (Start)

Hochstadt defines the standard Hermite polynomials of A066325 via H(x,n+1)=x*H(x,n)-n*H(x,n-1); note the index shift relative to the definition in the current sequence.

As a consequence, the polynomials defined here are orthogonal with weight exp(-x^2/2) in a restricted sense than the usual Hermite Polynomials, i.e. the integral of P(x,n)*P(x,m)*exp(-x^2/2) over x=-infinity..infinity vanishes for m=n-1 (mod 2), as for any system of polynomials with separated even and odd functions, but not for the general case of m<>n as with the Hermite polynomials H(x,n) or other classical polynomials. (End)

These polynomials appear in the radial l=0 (s) wave functions of the isotropic three-dimensional harmonic quantum oscillator with the dimensionless variable x=(r/L)^2 with r>=0 and L^2=h/(m*f0). h is Planck's constant and m and f0 are the mass and the frequency of the oscillator.

From Tom Copeland, Dec 13 2015: (Start)

See A099174 for relations to the Hermite polynomials and the link in A176230 for operator relations. The infinitesimal generator for this matrix contains A014105.

The row polynomials are P(n,x) = 2^n n! Lag(n,x/2,1/2), where Lag(n,x,q) is the associated Laguerre polynomial of order q, with raising operator R = -x^(-2) [x^(3/2) (1 - 2D)]^2 = 3 - x + (4x - 6) D - 4x D^2 with D = d/dx, i.e., R P(n,x) - P(n+1,x). A matrix reresentation of R acting on an o.g.f. (formal power series) is given by the transpose of the production matrix below. The diagonal corresponds to (3 + 4 xD) x^n = (3 + 4n) x^n; the upper diagonal, to -x x^n = -x^(n+1); and the lower diagonal, to (-6 - 4 xD) D x^n = -n (6 + 4(n-1)) x^(n-1), the sequence A002943. See A176230 for a similar relation.

The triangles of Bessel numbers entries A122848, A049403, A096713, A104556 contain these polynomials as even or odd rows. Also the aerated version A099174 and A066325. Reversed, these entries are A100861, A144299, A111924.

(End)

Exponential Riordan array [1/(1-2x)^(3/2), -x/(1-2x)]. - Paul Barry, Mar 07 2017

x^n = (1/2^n) * Sum_{k=0..n} a(n,k)*H_k(x).

These polynomials, H_n(x), are an Appell sequence, whose umbral compositional inverse sequence HI_n(x) consists of the same polynomials signed with the e.g.f. e^{-t^2} e^{xt}. Consequently, under umbral composition H_n(HI.(x)) = x^n = HI_n(H.(x)). Other differently scaled families of Hermite polynomials are A066325, A099174, and A060821. See Griffin et al. for a relation to the Catalan numbers and matrix integration. - Tom Copeland, Dec 27 2020

This irregular triangular array contains the integer coefficients of the normal ordering of the expansions of the differential operator R = (x + D)^n with D = d/dx. This operator is the raising/creation operator for the unsigned, modified Hermite polynomials H_n(x) of A099174; i.e., R H_n(x) = H_{n+1}(x). The lowering/annihilation operator L is D; i.e, L H_n(x) = D H_n(x) = n H_{n-1}(x).

Generalizing to the ladder operators for S_n(x) a general Sheffer polynomial sequence, (L + R)^n 1 = H_n(S.(x)), where the umbral composition is defined by H_n(S.(x)) = (h. + S.(x))^n = Sum_{k = 0..n} binomial(n,k) h_{n-k} S_k(x), where h_n are the Taylor series coefficients of e^{t^2/2}; i.e., e^{h.t} = e^{t^2/2}.

Another interpretation is that the n-th row contains the coefficients of the terms in the normal ordering of the 2^n permutations of two binary symbols given by (L + R)^n under the Leibniz condition LR = RL + 1 equivalent to the commutator relation [L,R] = LR - RL = 1. For example, (x + D)^2 = xx + xD + Dx + DD = x^2 + 2 xD + 1 + D^2.

As in the examples, the coefficients are ordered as those for the terms x^k D^m, ordered first from higher k to lower k and subordinately from lower m to higher m.

The number sequences associated to these polynomials are intimately related to the complete graphs K_n, which have n vertices and (n-1) edges. H_n(x) are the independence polynomials for K_n. The moments, h_n, of the H_n(x) Appell polynomials are the aerated double factorials A001147, the number of perfect matchings in the complete graph K_{n}, zero for odd n. The row lengths, A002620, give the number of maximal strokes on the complete graph K_n. The triangular numbers--the sum of two consecutive row lengths--give the number of edges of K_n. The row sums, A005425, are the number of matchings of the corona K'_n of the complete graph K_n and the complete graph K_1. K_n can be viewed as the projection onto a plane of the edges of the regular (n-1)-dimensional simplex, whose face polynomials are (x+1)^n - 1 (cf. A135278 and A074909).

The coefficients represent the combinatorics of a switchboard problem in which among n subscribers, a subscriber may talk to one of the others, someone on an outside line, or not at all--no conference calls allowed. E.g., the coefficient 12 for H_4(x+y) is the number of ways among 4 subscribers that a pair of subscribers can talk to each other while another is on an outside line and the remaining subscriber is disconnected. The coefficient 3 for the polynomial corresponds to the number of ways two pairs of subscribers can talk among themselves. This can be regarded as a dinner table problem also where a diner may exchange positions with another diner; remain seated; or get up, change plans, and sit back down in the same chair--no more than one exchange per diner.

From Peter Luschny, May 28 2021: (Start)

If we write the monomials of the row polynomials in degree-lexicographic order, we observe: The coefficient triangle appears as a series of concatenated subtriangles. The first one is Pascal's triangle A007318. Appending the rows of triangle A094305 begins in row 2. In row 4, the next triangle starts, which is A344565. This scheme seems to go on indefinitely. [This is now set out in A344911.] (End)

From Tom Copeland, May 29 2021: (Start)

Simplex model: H_n(x+y) = (h. + x + y)^n with h_n the aerated odd double factorials (h_0 = 1, h_1 = 0, h_2 = 1, h_3 = 0, h_4 = 3, h_5 = 0, h_6 = 15, ...), the number of perfect matchings for the n vertices of the (n-1)-Dimensional simplices, i.e., the hypertriangles, or hypertetrahedrons. This multinomial enumerates permutations of three families of objects--vertices labeled with either x or y, or perfect (pair) matchings of the unlabeled vertices for the n vertices of the (n-1)-D simplex. For example, (x+D)^2 = xx + xD + Dx + D^2 = x^2 + 2xD + D^2 + 1 corresponds to H_2(x+y) = (h.+ x + y)^2 = h_0 (x+y)^2 + 2 h_1 (x+y) + h_2 = x^2 + 2xy + y^2 + 1, which, in turn, corresponds to a line segment, the 1-D simplex, with both vertices labeled with x's; or one with an x, the other a y; or both vertices labeled with y's; or one unlabeled matched pair. The exponent k of x^k y^m represents the number of these vertices to which an x is assigned, and m, those with a y assigned. The remaining unlabeled vertices (a sub-simplex) form an independent edge set of (single color) colored edges, i.e., no colored edge is touching another colored edge (a perfect matching for the sub-simplex) nor the vertices assigned an x or y. Accordingly, the coefficients for k=m=0 represent the number of ways to assign a perfect matching to the simplex--zero for simplices with an odd number of vertices and the odd double factorials (1, 3, 15, 105, ...) for the simplices with an even number. For the simplices with an odd number of vertices, the coefficients for k+m=1 are the odd double factorials. (Note the polynomials are invariant upon interchange of the variables x and y.) (End)

Row sums are A066223. Reverse of A119743. Inverse is alternating sign version.

Diagonal sums are essentially A025164.

From Tom Copeland, Dec 13 2015: (Start)

See A099174 for relations to the Hermite polynomials and the link for operator relations, including the infinitesimal generator containing A000384.

Row polynomials are 2^n n! Lag(n,-x/2,-1/2), where Lag(n,x,q) is the associated Laguerre polynomial of order q.

The triangles of Bessel numbers entries A122848, A049403, A096713, A104556 contain these polynomials as even or odd rows. Also the aerated version A099174 and A066325. Reversed, these entries are A100861, A144299, A111924.

Divided along the diagonals by the initial element (A001147) of the diagonal, this matrix becomes the even rows of A034839.

(End)

The first few rows appear in expansions related to the Dedekind eta function on pp. 537-538 of the Chan et al. link. - Tom Copeland, Dec 14 2016

Triangle related to A000085.

Riordan array [exp(x(2+x)/2),x]. - Paul Barry, Nov 05 2008

Array is exp(S+S^2/2) where S is A132440 the infinitesimal generator for Pascal's triangle. T(n,k) gives the number of ways to choose a subset of {1,2,...,n} of size k and then partitioning the remaining n-k elements into sets each of size 1 or 2. Cf. A122832. - Peter Bala, May 14 2012

T(n,k) is equal to the number of R-classes (equivalently, L-classes) in the D-class consisting of all rank k elements of the partial Brauer monoid of degree n. - James East, Aug 17 2015