A115326 - OEIS

%I #19 Jan 30 2020 21:29:15

%S 1,1,9,49,625,6561,109561,1697809,35247969,717436225,17862589801,

%T 448030761201,13029739166929,387070092765409,12888060720104025,

%U 441427773256896721,16566268858818121921,641658452161285040769

%N E.g.f.: exp(x/(1-2*x))/sqrt(1-4*x^2).

%C Term-by-term square of sequence with e.g.f.: exp(x+m/2*x^2) is given by e.g.f.: exp(x/(1-m*x))/sqrt(1-m^2*x^2) for all m.

%H Vincenzo Librandi, <a href="/A115326/b115326.txt">Table of n, a(n) for n = 0..200</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HermitePolynomial.html">Hermite Polynomial</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Hermite_polynomials">Hermite polynomials</a>

%F Equals the term-by-term square of A047974 which has e.g.f.: exp(x+x^2).

%F D-finite with recurrence: a(n) = (2*n-1)*a(n-1) + 2*(n-1)*(2*n-1)*a(n-2) - 8*(n-2)^2*(n-1)*a(n-3). - _Vaclav Kotesovec_, Sep 25 2013

%F a(n) ~ 2^(n-1)*n^n*exp(sqrt(2*n)-n-1/4) * (1 + 13/(24*sqrt(2*n))). - _Vaclav Kotesovec_, Sep 25 2013

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

%t CoefficientList[Series[E^(x/(1-2*x))/Sqrt[1-4*x^2], {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Sep 25 2013 *)

%t RecurrenceTable[{a[0] == a[1] == 1, a[2] == 9, a[n] == (2 n - 1) a[n - 1] + 2 (n - 1) (2 n - 1) a[n - 2] - 8 (n - 2)^2 (n - 1) a[n - 3]}, a, {n, 20}] (* _Bruno Berselli_, Sep 27 2013 *)

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

%o (PARI) a(n)=local(m=2);n!*polcoeff(exp(x/(1-m*x+x*O(x^n)))/sqrt(1-m^2*x^2+x*O(x^n)),n)

%Y Cf. A047974.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jan 20 2006