A111882 Row sums of triangle A111595 (normalized rescaled squared Hermite polynomials).

1, 1, 0, 4, 4, 36, 256, 400, 17424, 784, 1478656, 876096, 154753600, 560363584, 19057250304, 220388935936, 2564046397696, 83038749753600, 327933273309184, 33173161139160064, 26222822450021376, 14475245839622726656

Offset: 0

Author

Wolfdieter Lang, Aug 23 2005

Keywords

• nonn
• easy

Links

• G. C. Greubel, Table of n, a(n) for n = 0..449

Crossrefs

Cf. A111883 (unsigned row sums of A111595).

Programs

Magma

m:=30; R:=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 10 2018

Mathematica

With[{nmax = 50}, CoefficientList[Series[Exp[x/(1 + x)]/Sqrt[1 - x^2], {x, 0, nmax}], x]*Range[0, nmax]!] (* G. C. Greubel, Jun 10 2018 *)

PARI

x='x+O('x^30); Vec(serlaplace(exp(x/(1+x))/sqrt(1-x^2))) \\ G. C. Greubel, Jun 10 2018

Python

from sympy import hermite, Poly, sqrt
def a(n): return sum(Poly(1/2**n*hermite(n, sqrt(x/2))**2, x).all_coeffs()) # Indranil Ghosh, May 26 2017

Formula

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

a(n) = Sum_{m=0..n} A111595(n, m), n>=0.

A111882(n) = A001464(n)^2. - Mark van Hoeij, Nov 11 2009

D-finite with recurrence a(n) +(n-2)*a(n-1) -(n-1)*(n-2)*a(n-2) -(n-1)*(n-2)^2*a(n-3)=0. - R. J. Mathar, Oct 05 2014