A199577 -id:A199577 - cp's OEIS frontend

A002793 a(n) = 2na(n-1) - (n-1)^2a(n-2).

0, 1, 4, 20, 124, 920, 7940, 78040, 859580, 10477880, 139931620, 2030707640, 31805257340, 534514790680, 9591325648580, 182974870484120, 3697147584561340, 78861451031150840, 1770536585183202980, 41729280102868841080, 1030007496863617367420, 26568602827124392999640

Offset: 0

N. J. A. Sloane, Robert G. Wilson v

nonn

From Wolfdieter Lang, Dec 12 2011: (Start)
r(n) = a(n+1)*(-1)^n, n >= 0, gives the alternating row sums of the coefficient triangle A199577, i.e., r(n)=La_n(1;0,-1), with the monic first associated Laguerre polynomials with parameter alpha=0 evaluated at x=-1.
The e.g.f. for these row sums r(n) is g(x) = -(2+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(1,x) = Gamma(0,x).
This e.g.f. satisfies the homogeneous ordinary second-order differential equation (1+x)^2*(d^2/dx^2)g(x) + (6+5*x)*(d/dx)g(x) + 4*g(x) = 0, g(0)=1, (d/dx)g(x)|_{x=0}=-4.
This e.g.f. g(x) is equivalent to the recurrence
  b(n)= -2*(n+1)*b(n-1) - n^2*b(n-2), b(-1)=0, b(0)=1.
Therefore, the e.g.f. of a(n) is A(x)=int(g(-x),x), with A(0)=0. This agrees with the e.g.f. given below in the formula section by Max Alekseyev.
(End)

J. Ser, Les Calculs Formels des Séries de Factorielles. Gauthier-Villars, Paris, 1933, p. 78.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
H. S. Wall, Analytic Theory of Continued Fractions, Chelsea 1973, p. 356.

G. C. Greubel, Table of n, a(n) for n = 0..440
Emanuele Munarini, Shifting Property for Riordan, Sheffer and Connection Constants Matrices, Journal of Integer Sequences, Vol. 20 (2017), Article 17.8.2.
J. Ser, Les Calculs Formels des Séries de Factorielles, Gauthier-Villars, Paris, 1933 [Local copy].
J. Ser, Les Calculs Formels des Séries de Factorielles (Annotated scans of some selected pages)

Bisection of A056952. A199577 (alternating row sums, unsigned).

Cf. A002720, A073003.

I:=[1, 4]; [0] cat [n le 2 select I[n] else 2*n*Self(n-1) - (n-1)^2*Self(n-2): n in [1..30]]; // G. C. Greubel, May 16 2018

Flatten[{0,RecurrenceTable[{(-1+n)^2 a[-2+n]-2 n a[-1+n]+a[n]==0,a[1]==1,a[2]==4}, a, {n, 20}]}] (* Vaclav Kotesovec, Oct 19 2013 *)
nxt[{n_,a_,b_}]:={n+1,b,2(n+1)b-n^2 a}; NestList[nxt,{1,0,1},30][[All,2]] (* Harvey P. Dale, Sep 06 2022 *)

A058006(n) = sum(k=0,n, (-1)^k*k! );
a(n) = if (n<=1, n, sum(k=1, n, (k+1) * A058006(k-1) * binomial(n,k) * (n-1)! / (k-1)! ) ); /* Joerg Arndt, Oct 12 2012 */

{a(n)=if(n==1,1,polcoeff(1-sum(m=1, n-1, a(m)*x^m*(1-(m+1)*x+x*O(x^n))^2), n))} \\ Paul D. Hanna, Feb 06 2013

From Max Alekseyev, Jul 06 2010: (Start)
For n > 1, a(n) = Sum_{k=1..n} (k+1) * A058006(k-1) * binomial(n,k) * (n-1)! / (k-1)!.
E.g.f.: (Gamma(0,1) - Gamma(0,1/(1-x))) * exp(1/(1-x)) / (1-x). (End)
From Peter Bala, Oct 11 2012: (Start)
Numerators in the sequence of convergents of Stieltjes's continued fraction for A073003, the Euler-Gompertz constant G := int {x = 0..oo} 1/(1+x)*exp(-x) dx:
G = 1/(2 - 1^2/(4 - 2^2/(6 - 3^2/(8 - ...)))). See [Wall, Chapter 18, (92.7) with a = 1]. The sequence of convergents to the continued fraction begins [1/2, 4/7, 20/34, 124/209, ...]. The denominators are in A002720.
(End)
G.f.: x = Sum_{n>=1} a(n) * x^n * (1 - (n+1)*x)^2. - Paul D. Hanna, Feb 06 2013
a(n) ~ G * exp(2*sqrt(n) - n - 1/2) * n^(n+1/4) / sqrt(2) * (1 + 31/(48*sqrt(n))), where G = 0.596347362323194... is the Gompertz constant (see A073003). - Vaclav Kotesovec, Oct 19 2013

Edited by Max Alekseyev, Jul 13 2010

A201199 Triangle version of the array w(N,L) of the total number of round trips of length L on closed Laguerre graphs Lc_N.

Original entry on oeis.org

1, 1, 2, 1, 4, 3, 1, 18, 9, 4, 1, 76, 53, 16, 5, 1, 322, 357, 120, 25, 6, 1, 1364, 2489, 1024, 233, 36, 7, 1, 5778, 17509, 9424, 2545, 404, 49, 8, 1, 24476, 123449, 89536, 29985, 5400, 645, 64, 9, 1, 103682, 870893, 862560, 367505, 78392, 10213, 968, 81, 10

Offset: 0

Wolfdieter Lang, Nov 30 2011

For Laguerre graphs (open and closed ones) see the W. Lang link on Jacobi graphs under A201198. There one also finds a sketch of the closed Laguerre graph Lc_4 as Fig.4.

The total number of round trips on the closed Laguerre graph Lc_N, for N>=3, with N vertices N^2 loops, binomial(N,2) lines between neighboring vertices and two lines between the first and the last vertex (in total (3*N-1)*N/2+2 = (3*N^2-N+4)/2 lines) is w(N,L) = sum(w(N,L;p_n->p_n),n=1..N) = Trace((L_N)^L) = sum((x_n^{(N)})^L,n=1..N), with the N x N symmetric adjacency matrix, also called Lc_N, having non-vanishing elements (Lc_N)[n,n] = 2*n-1, n=1..N, (Lc_N)[n,n+1] = (Lc_N)[n+1,n] = n, n=1..N-1, and (Lc_N)[1,N]= 2=(Lc_N)[N,1]. The eigenvalues of Lc_N are x_n^{(N)}. They are the zeros of the characteristic polynomial Lac_N(x):=Det(x*1_N -Lc_N) with the N x N unit matrix 1_N. These are the polynomials Lac_N(x) = La(N,x) - 4*La1(N-2,x) - 4*(N-1)!, with the ordinary monic Laguerre polynomials La(N,x) with coefficient array given by A021009(n,m)*(-1)^n and the first associated monic Laguerre polynomials La1(N-2,x) with coefficient array given by A199577(n,m). For N=1 one has Lc_1=L_1 (Laguerre graph with one vertex and one loop) with L_1(x)=x-1, and for N=2 one has a graph where one vertex has one loop, the other three, and there are two lines joining these vertices, hence Lc_2(x)= x^2-4*x-1.

			The array w(N,L) starts:
N\L 0   1    2     3      4        5         6  ...
1:  1   1    1     1      1        1         1
2:  2   4   12    40    136      464      1584
3:  3   9   53   357   2489    17509    123449
4:  4  16  120  1024   9424    89536    862560
5:  5  25  233  2545  29985   367505   4599521
6:  6  36  404  5400  78392  1188336  18460016
7:  7  49  645 10213 176473  3195829  59473593
8:  8  64  968 17728 355536  7493504 162671840
9:  9  81 1385 28809 657953 15826041 392792273
...The triangle a(K,N) = w(N,K-N+1) starts:
K\N 1      2       3      4      5     6     7   8  9..
0:  1
1:  1      2
2:  1      4       3
3:  1     18       9      4
4:  1     76      53     16      5
5:  1    322     357    120     25     6
6:  1   1364    2489   1024    233    36     7
7:  1   5778   17509   9424   2545   404    49   8
8:  1  24476  123449  89536  29985  5400   645  64  9
...
For the graph Lc_4, shown in the W. Lang link as Figure 4, the counting for round trips of length L=2 for each of the four vertices V_i, i=1..4, read from left to right, is as follows.
V_1: 1+1+(1+1+2*1), V_2: 3+2*binomial(3,2)+1+(1+1+2*1),
V_3: 5+2*binomial(5,2)+(1+1+2*1)+(3+2*binomial(3,2)),
V_4: 7+2*binomial(7,2)+(3+2*binomial(3,2))+(1+1+2*1),
this sums to the total number  w(4,2)= 120  =  a(5,4).
Compared to the open L_4 graph (see the corresponding A201198 entry 4*28 = 112) one has to add 2*(1+1+2*1)=8 from the new two lines joining V_1 and V_4.

Wolfdieter Lang, Counting walks on Jacobi graphs: an application of orthogonal polynomials.

Cf. A201198 (open Laguerre graphs).

a(K,N) =  w(N,K-N+1),K>=0, N=1,...,K+1, with w(N,L) the total number of round trips of length L on the closed Laguerre graph Lc_N described above in the comment section.
The o.g.f. of w(N,L) is: G(N,x)=y*(d/dx)Lac_N(x)/Lac_N(x) with y=1/x.
  The characteristic polynomial Lac_N(x) has also been given in the comment section above.

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).

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

Original entry on oeis.org

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

Offset: 0

Wolfdieter Lang, Nov 25 2011

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

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

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

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 *)

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

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

cp's OEIS Frontend

A002793 a(n) = 2n*a(n-1) - (n-1)^2*a(n-2).

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A201199 Triangle version of the array w(N,L) of the total number of round trips of length L on closed Laguerre graphs Lc_N.

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

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

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A002793 a(n) = 2na(n-1) - (n-1)^2a(n-2).