A201203 Alternating row sums of triangle A201201: first associated monic Laguerre-Sonin(e) polynomials with parameter alpha=1 evaluated at x=-1.

1, -5, 29, -201, 1631, -15173, 159093, -1854893, 23788271, -332613321, 5033396573, -81929955953, 1426898945343, -26468817431501, 520884561854501, -10836674357638293, 237603001692915983, -5475288709200573713, 132276033079845108621

Offset

0,2

Links

G. C. Greubel, Table of n, a(n) for n = 0..440

Michael Wallner, A bijection of plane increasing trees with relaxed binary trees of right height at most one, arXiv:1706.07163 [math.CO], 2017, Table 2 on p. 13.

Formula

a(n) = Sum_{k=0..n} ((-1)^k)*A201201(n,k), n>=0.

a(n)+(2*n+3)*a(n-1)+n*(n+1)*a(n-2)=0, a(-1)=0, a(0)=1. - R. J. Mathar, Dec 07 2011

From Wolfdieter Lang, Dec 11 2011: (Start)

E.g.f. from A201201 with x=-1, z->x: g(x) = exp(1/(1+x))*(3+2*x)*(exp(-1) + (Ei(1,1/(1+x))-Ei(1,1)))/(1+x)^4-(2+x)/(1+x)^3, with the exponential integral Ei.

This e.g.f. satisfies the homogeneous ordinary second-order differential equation (1+x)^2*(d^2(g(x))/dx^2) + (7+6*x)*(d(g(x))/dx)+6*g(x), with g(0)=1 and (d(g(x))/dx)_{x=0} = -5. This is equivalent to the recurrence conjectured above by R. J. Mathar, which is thus proved.

(End)

Let G denote Gompertz's constant A073003. The unsigned sequence is the sequence of numerators in the convergents coming from the infinite continued fraction expansion 1 - G = 1/(3 - 2/(5 - 6/(7 - ... - n*(n+1)/((2*n+3) - ...)))). The sequence of convergents begins [1/3, 5/13, 29/73, 201/501, ...]. The denominators are in A000262. - Peter Bala, Aug 19 2013

a(n) ~ (-1)^n * 2^(-1/2)*(exp(-1)-Ei(1,1)) * exp(2*sqrt(n)-n+1/2) * n^(n+7/4) * (1+91/(48*sqrt(n))), where Ei(1,1) = 0.21938393439552... = G / exp(1), where G = 0.596347362323194... is the Gompertz constant (see A073003). - Vaclav Kotesovec, Oct 19 2013

Maple

A201203 := proc(n)
    add((-1)^k*A201201(n, k), k=0..n) ;
end proc:
seq(A201203(n), n=0..20) ; # R. J. Mathar, Dec 07 2011

Mathematica

Flatten[{1, RecurrenceTable[{n*(1+n)*a[-2+n]+(3+2*n)*a[-1+n] +a[n]==0, a[1]==-5, a[2]==29}, a, {n, 20}]}] (* Vaclav Kotesovec, Oct 19 2013 *)

Crossrefs

Cf. A201201, A201202 (row sums), A073003, A002793.

Sequence in context: A367050 A216313 A108453 * A004213 A350476 A105277

Adjacent sequences: A201200 A201201 A201202 * A201204 A201205 A201206

Keyword

sign,easy

Author

Wolfdieter Lang, Dec 06 2011

Extensions

R. J. Mathar conjecture corrected and proved by Wolfdieter Lang, Dec 11 2011

Status

approved