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Row sums of signed triangle A062139 (generalized a=2 Laguerre).
2

%I #28 Sep 08 2022 08:45:03

%S 1,2,5,14,37,34,-887,-14050,-168919,-1916542,-21607859,-245387858,

%T -2799384755,-31558843486,-337767590383,-3063846770626,

%U -11912361112367,477367592119810,21032925955607701,627398853149961038,16703816669710968821

%N Row sums of signed triangle A062139 (generalized a=2 Laguerre).

%H G. C. Greubel, <a href="/A062197/b062197.txt">Table of n, a(n) for n = 0..448</a>

%H <a href="/index/La#Laguerre">Index entries for sequences related to Laguerre polynomials</a>

%F E.g.f.: exp(-x/(1-x))/(1-x)^3.

%F a(n) = Sum_{m=0..n} ((-1)^m)*n!*binomial(n+2, n-m)/m!.

%F a(n) = 2*n*a(n-1) - (n-1)*(n+1)*a(n-2). - _Vaclav Kotesovec_, Aug 01 2013

%F a(n) = (n+2)!*hypergeom([-n],[3],1)/2. - _Peter Luschny_, Apr 11 2015

%p a := n -> (n+2)!*hypergeom([-n],[3],1)/2:

%p seq(simplify(a(n)), n=0..20); # _Peter Luschny_, Apr 11 2015

%t Table[n!*LaguerreL[n, 2, 1],{n,0,20}] (* _Vaclav Kotesovec_, Aug 01 2013 *)

%o (PARI) for(n=0,30, print1(n!*sum(k=0,n, (-1)^k*binomial(n+2, n-k)/k!), ", ")) \\ _G. C. Greubel_, May 13 2018

%o (PARI) a(n) = vecsum(Vec(n!*pollaguerre(n, 2))); \\ _Michel Marcus_, Feb 06 2021

%o (Magma) [Factorial(n)*(&+[(-1)^k*Binomial(n+2, n-k)/Factorial(k): k in [0..n]]): n in [0..30]]; // _G. C. Greubel_, May 13 2018

%Y Cf. A062139.

%K sign,easy

%O 0,2

%A _Wolfdieter Lang_, Jun 19 2001