OFFSET
0,3
LINKS
Indranil Ghosh, Table of n, a(n) for n = 0..100
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
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 *)
PROG
(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
(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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Aug 23 2005
STATUS
approved