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A201202 Row sums of triangle A201201: first associated monic Laguerre polynomials with parameter alpha=1 evaluated at x=1.

%I A201202 #20 Dec 18 2017 22:47:17
%S A201202 1,-3,9,-27,63,117,-4167,55953,-651177,7336593,-82438983,927666333,
%T A201202 -10331176977,110106505773,-1023541502247,5304225184137,
%U A201202 103363857534663,-5240827920059127,158560193765332953,-4192332947225516907,105290369454806352927
%N A201202 Row sums of triangle A201201: first associated monic Laguerre polynomials with parameter alpha=1 evaluated at x=1.
%F A201202 a(n)=sum(A201201(n,k),k=0..n), n>=0.
%F A201202 Apparently a(n)+(2*n+1)*a(n-1)+n*(n+1)*a(n-2)=0, a(-1)=0, a(1)=1. - _R. J. Mathar_, Dec 07 2011
%F A201202 From _Wolfdieter Lang_, Dec 12 2011: (Start)
%F A201202 E.g.f. from A201201 with x=1, z->x: g(x)=(1+2*x)*exp(-1/(1+x))*(exp(1)-((Ei(1,-1/(1+x)) - Ei(1,-1))))/(1+x)^4 - x/(1+x)^3, 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 afterwards.
%F A201202 This e.g.f. satisfies the homogeneous second-order differential equation: (1+x)^2*(d^2/dx^2)g(x) + (5+6*x)*(d/dx)g(x) + 6*g(x) = 0, with g(0)=1 and (d/dx)g(x)|_{x=0} = -3. This is equivalent to the recurrence conjectured by _R. J. Mathar_, which is thus proved.
%F A201202 (End)
%p A201202 A201202 := proc(n)
%p A201202     add(A201201(n,k),k=0..n) ;
%p A201202 end proc:
%p A201202 seq(A201202(n),n=0..20) ; # _R. J. Mathar_, Dec 07 2011
%t A201202 a[n_, k_] := (-1)^(n-k)*((n+1)*(n+1)!/((k+1)*(k+1)!))*Binomial[n, k]*HypergeometricPFQ[{-(n-k), k, 1}, {-(n+1), k+2}, 1]; Table[Sum[a[n, k], {k, 0, n}], {n, 0, 20}]  (* _Jean-François Alcover_, Jun 21 2013 *)
%Y A201202 Cf. A201201, A201203 (alternating row sums).
%K A201202 sign,easy
%O A201202 0,2
%A A201202 _Wolfdieter Lang_, Dec 06 2011