A115326 E.g.f.: exp(x/(1-2*x))/sqrt(1-4*x^2).

1, 1, 9, 49, 625, 6561, 109561, 1697809, 35247969, 717436225, 17862589801, 448030761201, 13029739166929, 387070092765409, 12888060720104025, 441427773256896721, 16566268858818121921, 641658452161285040769

Offset: 0

Author

Paul D. Hanna, Jan 20 2006

Keywords

• nonn

Comments

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.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Eric Weisstein's World of Mathematics, Hermite Polynomial
• Wikipedia, Hermite polynomials

Crossrefs

Cf. A047974.

Programs

Mathematica

CoefficientList[Series[E^(x/(1-2*x))/Sqrt[1-4*x^2], {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Sep 25 2013 *)
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 *)
Table[Abs[HermiteH[n, I/2]]^2, {n, 0, 20}] (* Vladimir Reshetnikov, Oct 11 2016 *)

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)

Formula

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

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

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

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