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A199578 Row sums of coefficient triangle of the monic associated Laguerre polynomials of order 1.

%I A199578 #22 Sep 08 2022 08:46:00
%S A199578 1,-2,4,-6,-16,310,-3144,28826,-260000,2345094,-20901880,176084986,
%T A199578 -1216168944,1862029910,186232275544,-6005924996070,144514137334976,
%U A199578 -3177768345524954,67577079942366120,-1420754665075404166,29799354626069718640
%N A199578 Row sums of coefficient triangle of the monic associated Laguerre polynomials of order 1.
%H A199578 G. C. Greubel, <a href="/A199578/b199578.txt">Table of n, a(n) for n = 0..448</a>
%F A199578 a(n) = Sum_{k=0..n} A199577(n,k), n>=0.
%F A199578 From _Wolfdieter Lang_, Dec 12 2011 (Start)
%F A199578 E.g.f. from A199577 with x=1, z->x:
%F A199578 g(x) = -x*exp(-1/(1+x))*(Ei(1,-1/(1+x))-Ei(1,-1))/(1+x)^3 + 1/(1+x)^2, with the exponential integral Ei. In order to obtain the series use first Ei(1,-y/(1+x))-Ei(1,-y), and put y=1 after the expansion.
%F A199578 This e.g.f. satisfies the homogeneous ordinary second order differential equation (1+x)^2*(d^2/dx^2)g(x)+(4+5*x)*(d/dx)g(x)+4*g(x) = 0, with g(0)=1 and (d/dx)g(x)|_{x=0}=-2.
%F A199578 This e.g.f. is equivalent to the recurrence relation:
%F A199578 a(n) = -2*n*a(n-1) - n^2*a(n-2),  a(-1)=0, a(0)=1.
%F A199578 (End)
%F A199578 The conjecture on the alternating row sums has been proved by _Wolfdieter Lang_, Dec 12 2011
%t A199578 RecurrenceTable[{a[n] == -2*n*a[n-1] -n^2*a[n-2], a[0] == 1, a[1] == -2}, a, {n, 0, 40}] (* _G. C. Greubel_, May 14 2018 *)
%o A199578 (Magma) I:=[-2,4]; [1] cat [n le 2 select I[n] else -2*n*Self(n-1) - n^2*Self(n-2): n in [1..30]]; // _G. C. Greubel_, May 14 2018
%o A199578 (PARI) m=30; v=concat([-2,4], vector(m-2)); for(n=3, m, v[n]=-2*n*v[n-1]-n^2*v[n-2]); concat([1], v) \\ _G. C. Greubel_, May 14 2018
%Y A199578 Cf. A199577 (monic first associated Laguerre), A002793(n+1)*(-1)^n, n>=0 (alternating row sums).
%K A199578 sign,easy
%O A199578 0,2
%A A199578 _Wolfdieter Lang_, Nov 25 2011