A201202 -id:A201202 - cp's OEIS frontend

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

Wolfdieter Lang, Dec 06 2011

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.

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

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

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 *)

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

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

A201201 Coefficient triangle for the monic associated Laguerre-Sonin(e) polynomials of order one for parameter alpha=1.

Original entry on oeis.org

1, -4, 1, 18, -10, 1, -96, 86, -18, 1, 600, -756, 246, -28, 1, -4320, 7092, -3168, 552, -40, 1, 35280, -71856, 41112, -9720, 1070, -54, 1, -322560, 787824, -552240, 165720, -24600, 1878, -70, 1, 3265920, -9329760, 7768080, -2835360, 531480, -54516, 3066, -88

Offset: 0

Wolfdieter Lang, Dec 06 2011

See A199577 for general comments on associated Laguerre-Sonin(e) polynomials of order m, and the Ismail reference.
The monic row polynomials are La_n(1;1,x) = sum(a(n,k)*x^k,k=0..n), with the three term recurrence
La_n(1;1,x) = (x-2*(n+1))*La_{n-1}{1;1,x) - n*(n+1)*La_{n-2}{1;1,x), La_{-1}{1;1,x)=0, La_0(1;1,x)=1.
In the Ismail reference the non-monic associated Laguerre polynomials of order 1 appear on p. 160 in Theorem 5.6.1, eq. 5.6.11. The connection is: La_n(1;1,x)= L_n^{(alpha=1)}(x;1)*(n+1)!*(-1)^n.
The e.g.f. gLa(1;1,z,x) for La_n(1;1,x) can be obtained from the o.g.f. G(1;1,z,x) for the non-monic version L_n^{alpha=1}(x;1) by gLa(1;1,z,x)= (d/dz) (z*G(1;1,-z,x)).
G(1;1,z,x) satisfies an ordinary first order inhomogeneous differential equation:
(d/dz) G(1;1,z,x) = (3/(1-z)+(1-x)/(1-z)^2-1/(z*(1-z)^2  z)^2))* G(1;1,z,x) + 1/(z*(1-z)^2), with G(1;1,z=0,x)=1. The standard solution is:
  G(1;1,z,x) = (exp(-x*z/(1-z))-1+z-x*exp(-x/(1-z))* (Ei(1,-x/(1-z))-Ei(1,-x)))/(z*(1-z)^2), with the exponential integral Ei(1,y)=int(exp(-t)/t,t=y..infty).

			The triangle begins:
n\k     0      1       2      3      4    5   6 7 ...
0:      1
1:     -4      1
2:     18    -10       1
3:    -96     86     -18      1
4:    600   -756     246    -28      1
5:  -4320   7092   -3168    552    -40    1
6:  35280 -71856   41112  -9720   1070  -54   1
7:-322560 787824 -552240 165720 -24600 1878 -70 1
...

M. E. H. Ismail (two chapters by W. Van Assche), Classical and Quantum Orthogonal Polynomials in One Variable, Cambridge University Press, 2005.

Cf. A199577 (alpha=0 case), A201202 (row sums), A201203 (alternating row sums).

La := proc(n,x)
    option remember;
    if n= -1 then
        0;
    elif n = 0 then
        1;
    else
        (x-2*n-2)*procname(n-1,x)-n*(n+1)*procname(n-2,x) ;
    end if;
end proc:
A201201 := proc(n,k)
    coeftayl( La(n,x),x=0,k) ;
end proc:
seq(seq(A201201(n,k),k=0..n),n=0..12) ; # R. J. Mathar, Dec 07 2011

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[a[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 21 2013 *)

a(n,k)=[x^k] La_n(1;1,x), n>=0, k=0,...,n.
a(n,k)= (-1)^(n-k)*((n+1)*(n+1)!/((k+1)*(k+1)!))*
binomial(n,k)*hypergeom(-(n-k),k,1; -(n+1),k+2; 1), 0<=k<=n. [Ismail, p. 161, eq. (5.6.18)].
The e.g.f. gLa(1;1,z,x) for La_n(1;1,x) is exp(-x/(1+z))*x*(x-2*(1+z))*(Ei(1,-x/(1+z)) - Ei(1,-x))/(1+z)^4 + exp(x*z/(1+z))*(-x+2*(1+z))/(1+z)^4 +(1+z+x)/(1+z)^3 -2/(1+z)^2, with the exponential integral Ei.
The e.g.f. gLa(1;1,z,x) for the Euler-derivative
  x*(d/dx) La_n(1;1,x) is x*exp(-x/(1+z))*(2*(1+z)-x)*
(Ei(1,-x/(1+z)) - Ei(1,-x))/(1+z)^4 + (1+z-x)*(1-exp(x*z/(1+z)))/(1+z)^3.
From this follows La_n(1;1,x) = (n+1)*La_n(1,x) -
  x*(d/dx)La_n(1;0,x). For La_n(1;0,x) see A199577 where it is called La_n(1;x).

cp's OEIS Frontend

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

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