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A099174 Triangle read by rows: coefficients of modified Hermite polynomials. 12
1, 0, 1, 1, 0, 1, 0, 3, 0, 1, 3, 0, 6, 0, 1, 0, 15, 0, 10, 0, 1, 15, 0, 45, 0, 15, 0, 1, 0, 105, 0, 105, 0, 21, 0, 1, 105, 0, 420, 0, 210, 0, 28, 0, 1, 0, 945, 0, 1260, 0, 378, 0, 36, 0, 1, 945, 0, 4725, 0, 3150, 0, 630, 0, 45, 0, 1, 0, 10395, 0, 17325, 0, 6930, 0, 990, 0, 55 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,8

COMMENTS

Absolute values of A066325.

T(n,k) is the number of involutions of {1,2,...,n}, having k fixed points (0<=k<=n). Example: T(4,2)=6 because we have 1243,1432,1324,4231,3214 and 2134. - Emeric Deutsch, Oct 14 2006

Riordan array [exp(x^2/2),x]. - Paul Barry, Nov 06 2008

Same as triangle of Bessel numbers of second kind, B(n,k) (see Cheon et al., 2013). - N. J. A. Sloane, Sep 03 2013

The modified Hermite polynomial h(n,x) (as in the Formula section) is the numerator of the rational function given by f(n,x) = x + (n - 2)/f(n-1,x), where f(x,0) = 1. - Clark Kimberling, Oct 20 2014

Second lower diagonal T(n,n-2) equals positive triangular numbers A000217 \ {0}. - M. F. Hasler, Oct 23 2014

From James East, Aug 17 2015: (Start)

T(n,k) is the number of R-classes (equivalently, L-classes) in the D-class consisting of all rank k elements of the Brauer monoid of degree n.

For n<k with n==k (mod 2), T(n,k) is the rank (minimal size of a generating set) and idempotent rank (minimal size of an idempotent generating set) of the ideal consisting of all rank <=k elements of the Brauer monoid. (End)

This array provides the coefficients of a Laplace-dual sequence H(n,x) of the Dirac delta function, delta(x), and its derivatives, formed by taking the inverse Laplace transform of these modified Hermite polynomials. H(n,x) = h(n,D) delta(x) with h(n,x) as in the examples and the lowering and raising operators L = -x and R = -x + D = -x + d/dx such that L H(n,x) = n * H(n-1,x) and R H(n,x) = H(n+1,x). The e.g.f. is exp[t H(.,x)] = e^(t^2/2) e^(t D) delta(x) = e^(t^2/2) delta(x+t). - Tom Copeland, Oct 02 2016

Antidiagonals of this entry are rows of A001497. - Tom Copeland, Oct 04 2016

LINKS

Table of n, a(n) for n=0..75.

P. Barry, Riordan array, orthogonal polynomials as moments, and Hankel transforms, arXiv:1102.0921 [math.CO], 2011.

G.-S. Cheon, J.-H. Jung and L. W. Shapiro, Generalized Bessel numbers and some combinatorial settings, Discrete Math., 313 (2013), 2127-2138.

James East and Robert D. Gray, Diagram monoids and Graham-Houghton graphs: idempotents and generating sets of ideals, arXiv:1404.2359 [math.GR], 2014. See Theorem 8.4 and Table 7. - James East, Aug 17 2015

A. Horzela, P. Blasiak, G. E. H. Duchamp, K. A. Penson and A. I. Solomon, A product formula and combinatorial field theory, arXiv:quant-ph/0409152, 2004.

Alexander Kreinin, Combinatorial Properties of Mills' Ratio, arXiv:1405.5852 [math.CO], 2014. See Table 2. - N. J. A. Sloane, May 29 2014

R. Paris, A uniform asymptotic expansion for the incomplete gamma function, Journal of Computational and Applied Mathematics, 148 (2002), p. 223-239 (See p. 329 and A137286. From Tom Copeland, Jan 03 2016).

R. Sazdanovic, A categorification of the polynomial ring, slide presentation, 2011

S. Yang and Z. Qiao, The Bessel numbers and Bessel matrices, Jrn. Math. Rsch. and Exposition, July 2011, Vol. 31, No. 4, pp.627-636. DOI:10.3770/j.issn:1000-341X.2011.04.006.

FORMULA

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

E.g.f.: exp(y*x + x^2/2). - Geoffrey Critzer, May 08 2012

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

T(n+2,n) = A000217(n+1), n >= 0. - M. F. Hasler, Oct 23 2014

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

From Tom Copeland, Dec 13 2015: (Start)

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 to n) binom(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 charactrization 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

EXAMPLE

h(0,x) = 1

h(1,x) = x

h(2,x) = x^2 + 1

h(3,x) = x^3 + 3*x

h(4,x) = x^4 + 6*x^2 + 3

h(5,x) = x^5 + 10*x^3 + 15*x

h(6,x) = x^6 + 15*x^4 + 45*x^2 + 15

From Paul Barry, Nov 06 2008: (Start)

Triangle begins

   1,

   0,  1,

   1,  0,  1,

   0,  3,  0,  1,

   3,  0,  6,  0,  1,

   0, 15,  0, 10,  0,  1,

  15,  0, 45,  0, 15,  0,  1

Production array starts

  0, 1,

  1, 0, 1,

  0, 2, 0, 1,

  0, 0, 3, 0, 1,

  0, 0, 0, 4, 0, 1,

  0, 0, 0, 0, 5, 0, 1 (End)

MAPLE

T:=proc(n, k) if n-k mod 2 = 0 then n!/2^((n-k)/2)/((n-k)/2)!/k! else 0 fi end: for n from 0 to 12 do seq(T(n, k), k=0..n) od; # yields sequence in triangular form; Emeric Deutsch, Oct 14 2006

MATHEMATICA

nn=10; a=y x+x^2/2!; Range[0, nn]!CoefficientList[Series[Exp[a], {x, 0, nn}], {x, y}]//Grid  (* Geoffrey Critzer, May 08 2102 *)

H[0, x_] = 1; H[1, x_] := x; H[n_, x_] := H[n, x] = x*H[n-1, x]-(n-1)* H[n-2, x]; Table[CoefficientList[H[n, x], x], {n, 0, 11}] // Flatten // Abs (* Jean-Fran├žois Alcover, May 23 2016 *)

PROG

(Sage)

def A099174_triangle(dim):

    M = matrix(SR, dim, dim)

    for n in (0..dim-1): M[n, n] = 1

    for n in (1..dim-1):

        for k in (0..n-1):

            M[n, k] = M[n-1, k-1]+(k+1)*M[n-1, k+1]

    return M

A099174_triangle(9)  # Peter Luschny, Oct 06 2012

(PARI) T(n, k)=if(k<=n && k==Mod(n, 2), n!/k!/(k=(n-k)/2)!>>k) \\ M. F. Hasler, Oct 23 2014

CROSSREFS

Row sums (polynomial values at x=1) are A000085.

Polynomial values: A005425 (x=2), A202834 (x=3), A202879(x=4).

Cf. A000217, A001147, A059343, A060821, A066325.

Cf. A000384. A014105, A034839, A049403, A096713, A100861, A104556, A122848, A130757, A176230, A176231.

Cf. A137286.

Cf. A001497.

Sequence in context: A256037 A179898 A066325 * A137297 A178117 A095710

Adjacent sequences:  A099171 A099172 A099173 * A099175 A099176 A099177

KEYWORD

nonn,tabl

AUTHOR

Ralf Stephan, on a suggestion of Karol A. Penson, Oct 13 2004

STATUS

approved

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Last modified March 26 12:52 EDT 2017. Contains 284126 sequences.