A111883 Unsigned row sums of triangle A111595 (normalized rescaled squared Hermite polynomials).

1, 1, 4, 16, 100, 676, 5776, 53824, 583696, 6864400, 90174016, 1274204416, 19642583104, 323196798016, 5714394630400, 107112895415296, 2135062451773696, 44858948563673344, 994634863541502976, 23133227941938073600, 564474119626559497216, 14388648533002088866816

Offset: 0

Author

Wolfdieter Lang, Aug 23 2005

Keywords

• nonn
• easy

Links

• Indranil Ghosh, Table of n, a(n) for n = 0..100

Crossrefs

Cf. A111882 (row sums of A111595).

Programs

Magma

m:=25; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(x/(1-x))/Sqrt(1-x^2))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Jun 09 2018

Mathematica

Table[Abs[HermiteH[n, I/Sqrt[2]]]^2/2^n, {n, 0, 20}] (* Vladimir Reshetnikov, Oct 11 2016 *)
CoefficientList[Series[Exp[t/(1-t)]/Sqrt[1-t^2],{t,0,100}],t] Range[0, 12]! (* Emanuele Munarini, Aug 31 2017 *)

PARI

a(n)=if(n<0, 0, n!*polcoeff(exp(x/(1-x)+x*O(x^n))/sqrt(1-x^2+x*O(x^n)),n)) /* Michael Somos, Aug 30 2005 */

Python

from sympy import hermite, Poly, sqrt, I
def a(n): return abs(Poly(hermite(n, I/sqrt(2)), x))**2/2**n # Indranil Ghosh, May 26 2017

Formula

E.g.f.: exp(x/(1-x))/sqrt(1-x^2).

a(n) = A000085(n)^2. - Michael Somos, Aug 30 2005

Conjecture: a(n) -n*a(n-1) -n*(n-1)*a(n-2) +(n-1)*(n-2)^2*a(n-3)=0. - R. J. Mathar, Oct 05 2014

Remark: the above conjectured recurrence is true and can be easily obtained by the e.g.f. - Emanuele Munarini, Aug 31 2017

a(n) = |H_n(i/sqrt(2))|^2 / 2^n = H_n(i/sqrt(2)) * H_n(-i/sqrt(2)) / 2^n, where H_n(x) is n-th Hermite polynomial, i = sqrt(-1). - Vladimir Reshetnikov, Oct 11 2016

a(n) ~ exp(2*sqrt(n) - n - 1/2) * n^n / 2. - Vaclav Kotesovec, Oct 01 2017