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Row sums of triangle A111595 (normalized rescaled squared Hermite polynomials).
3

%I #23 Sep 08 2022 08:45:21

%S 1,1,0,4,4,36,256,400,17424,784,1478656,876096,154753600,560363584,

%T 19057250304,220388935936,2564046397696,83038749753600,

%U 327933273309184,33173161139160064,26222822450021376,14475245839622726656

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

%H G. C. Greubel, <a href="/A111882/b111882.txt">Table of n, a(n) for n = 0..449</a>

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

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

%F A111882(n) = A001464(n)^2. - _Mark van Hoeij_, Nov 11 2009

%F Conjecture: 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

%t 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 *)

%o (Python)

%o from sympy import hermite, Poly, sqrt

%o def a(n): return sum(Poly(1/2**n*hermite(n, sqrt(x/2))**2, x).all_coeffs()) # _Indranil Ghosh_, May 26 2017

%o (PARI) x='x+O('x^30); Vec(serlaplace(exp(x/(1+x))/sqrt(1-x^2))) \\ _G. C. Greubel_, Jun 10 2018

%o (Magma) m:=30; 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 10 2018

%Y Cf. A111883 (unsigned row sums of A111595).

%K nonn,easy

%O 0,4

%A _Wolfdieter Lang_, Aug 23 2005