A062267 Row sums of (signed) triangle A060821 (Hermite polynomials).

1, 2, 2, -4, -20, -8, 184, 464, -1648, -10720, 8224, 230848, 280768, -4978816, -17257600, 104891648, 727511296, -1901510144, -28538404352, 11377556480, 1107214478336, 1759326697472, -42984354695168, -163379084079104

Offset: 0

Author

Wolfdieter Lang, Jun 19 2001

Keywords

• sign
• easy

Links

• G. C. Greubel, Table of n, a(n) for n = 0..730
• Index entries for sequences related to Hermite polynomials

Crossrefs

Cf. A000898, A121966.

Programs

Magma

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(x*(2-x)))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Jun 08 2018

Maple

A062267 := proc(n)
    HermiteH(n,1) ;
    simplify(%) ;
end proc: # R. J. Mathar, Feb 05 2013

Mathematica

lst={};Do[p=HermiteH[n,1];AppendTo[lst,p],{n,0,5!}];lst (* Vladimir Joseph Stephan Orlovsky, Jun 15 2009 *)
Table[2^n HypergeometricU[-n/2, 1/2, 1], {n, 0, 23}] (* Benedict W. J. Irwin, Oct 17 2017 *)
With[{nmax=50}, CoefficientList[Series[Exp[x*(2-x)], {x,0,nmax}],x]* Range[0, nmax]!] (* G. C. Greubel, Jun 08 2018 *)

PARI

x='x+O('x^30); Vec(serlaplace(exp(-x*(x-2)))) \\ G. C. Greubel, Jun 08 2018

PARI

a(n) = polhermite(n,1); \\ Michel Marcus, Jun 09 2018

Python

from sympy import hermite, Poly
def a(n): return sum(Poly(hermite(n, x), x).all_coeffs()) # Indranil Ghosh, May 26 2017

Formula

a(n) = Sum_{m=0..n} A060821(n, m) = H(n, 1), with the Hermite polynomials H(n, x).

E.g.f.: exp(-x*(x-2)).

a(n) = 2*(a(n - 1) - (n - 1)*a(n - 2)). - Roger L. Bagula, Sep 11 2006

a(n) = 2^n * U(-n/2, 1/2, 1), where U is the confluent hypergeometric function. - Benedict W. J. Irwin, Oct 17 2017

E.g.f.: Product_{k>=1} ((1 + x^k)/(1 - x^k))^(mu(k)/k). - Ilya Gutkovskiy, May 26 2019