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

1, -2, 4, -6, -16, 310, -3144, 28826, -260000, 2345094, -20901880, 176084986, -1216168944, 1862029910, 186232275544, -6005924996070, 144514137334976, -3177768345524954, 67577079942366120, -1420754665075404166, 29799354626069718640

Offset: 0

Author

Wolfdieter Lang, Nov 25 2011

Keywords

• sign
• easy

Links

• G. C. Greubel, Table of n, a(n) for n = 0..448

Crossrefs

Cf. A199577 (monic first associated Laguerre), A002793(n+1)*(-1)^n, n>=0 (alternating row sums).

Programs

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

Mathematica

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 *)

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

Formula

a(n) = Sum_{k=0..n} A199577(n,k), n>=0.

From Wolfdieter Lang, Dec 12 2011 (Start)

E.g.f. from A199577 with x=1, z->x:

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.

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.

This e.g.f. is equivalent to the recurrence relation:

a(n) = -2*n*a(n-1) - n^2*a(n-2), a(-1)=0, a(0)=1.

(End)

The conjecture on the alternating row sums has been proved by Wolfdieter Lang, Dec 12 2011