A277378 Expansion of e.g.f. exp(2*x/(1-x))/sqrt(1-x^2).

1, 2, 9, 50, 361, 3042, 29929, 331298, 4100625, 55777922, 828691369, 13316140818, 230256982201, 4257449540450, 83834039024649, 1750225301567618, 38614608429012001, 897325298084953602, 21904718673762721225, 560258287738117292018, 14981472258320814527241

Offset: 0

Author

Vladimir Reshetnikov, Oct 11 2016

Keywords

• nonn

Links

• Eric Weisstein's World of Mathematics, Hermite Polynomial.
• Wikipedia, Hermite polynomials.

Crossrefs

Cf. A000321, A000898, A059343, A062267, A067994, A277280, A277281.

Programs

Mathematica

Table[Abs[HermiteH[n, I]]^2/2^n, {n, 0, 20}]
With[{nn=20},CoefficientList[Series[Exp[2x/(1-x)]/Sqrt[1-x^2],{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, Jan 27 2023 *)

Formula

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

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

D-finite with recurrence: (n+2)*(a(n) + n*a(n-1)) = a(n+1) + n*(n-1)^2*a(n-2).

a(n) ~ n^n / (2 * exp(1 - 2*sqrt(2*n) + n)) * (1 + 2*sqrt(2)/(3*sqrt(n))). - Vaclav Kotesovec, Oct 27 2021