A111883 - OEIS

%I #27 Sep 08 2022 08:45:21

%S 1,1,4,16,100,676,5776,53824,583696,6864400,90174016,1274204416,

%T 19642583104,323196798016,5714394630400,107112895415296,

%U 2135062451773696,44858948563673344,994634863541502976,23133227941938073600,564474119626559497216,14388648533002088866816

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

%H Indranil Ghosh, <a href="/A111883/b111883.txt">Table of n, a(n) for n = 0..100</a>

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

%F a(n) = A000085(n)^2. - _Michael Somos_, Aug 30 2005

%F 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

%F Remark: the above conjectured recurrence is true and can be easily obtained by the e.g.f. - _Emanuele Munarini_, Aug 31 2017

%F 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

%F a(n) ~ exp(2*sqrt(n) - n - 1/2) * n^n / 2. - _Vaclav Kotesovec_, Oct 01 2017

%t Table[Abs[HermiteH[n, I/Sqrt[2]]]^2/2^n, {n, 0, 20}] (* _Vladimir Reshetnikov_, Oct 11 2016 *)

%t CoefficientList[Series[Exp[t/(1-t)]/Sqrt[1-t^2],{t,0,100}],t] Range[0, 12]! (* _Emanuele Munarini_, Aug 31 2017 *)

%o (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 */

%o (Python)

%o from sympy import hermite, Poly, sqrt, I

%o def a(n): return abs(Poly(hermite(n, I/sqrt(2)), x))**2/2**n # _Indranil Ghosh_, May 26 2017

%o (Magma) m:=25; R<x>:=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

%Y Cf. A111882 (row sums of A111595).

%K nonn,easy

%O 0,3

%A _Wolfdieter Lang_, Aug 23 2005