A021009 -id:A021009 - cp's OEIS frontend

A201198 Triangle version of the array w(N,L) of the average number of round trips of length L on Laguerre graphs L_N.

1, 1, 1, 1, 2, 1, 1, 6, 3, 1, 1, 20, 15, 4, 1, 1, 68, 87, 28, 5, 1, 1, 232, 531, 232, 45, 6, 1, 1, 792, 3303, 2056, 485, 66, 7, 1, 1, 2704, 20691, 18784, 5645, 876, 91, 8, 1, 1, 9232, 129951, 174112, 68245, 12636, 1435, 120, 9, 1

Offset: 0

Wolfdieter Lang, Nov 30 2011

For Laguerre graphs see the W. Lang link on Jacobi graphs (named after the symmetric tridiagonal Jacobi adjacency matrices, related to orthogonal polynomials). There one also finds a sketch of the Laguerre graph L_4 in Fig. 3.

The average number of round trips for the Laguerre graph L_N with N vertices, N^2 loops and binomial(N,2) lines between neighboring vertices (in total (3*N-1)*N/2 lines) is w(N,L) = sum(w(N,L;p_n->p_n), n=1..N)/N = Trace((L_N)^L)/N = sum((x_n^{(N)})^L,n = 1..N)/N, with the N x N tridiagonal symmetric adjacency matrix L_N, having non-vanishing elements (L_N)[n,n] = 2*n-1, n=1..N, (L_N)[n,n+1] = (L_N)[n+1,n] = n, n=1..N-1. The eigenvalues of L_N are x_n^{(N)}. They are the zeros of the characteristic polynomial La_N(x):=Det(x*1_N -L_N) with the N x N unit matrix 1_N. These are the ordinary monic Laguerre polynomials with coefficient triangle given in A021009(n,m)*(-1)^n.

			The array w(N,L) starts:
N\L 0 1   2          4       5        6          7 ...
1:  1 1   1    1     1       1        1          1
2:  1 2   6   20    68     232      792       2704
3:  1 3  15   87   531    3303    20691     129951
4:  1 4  28  232  2056   18784   174112    1625152
5:  1 5  45  485  5645   68245   841725   10495525
6:  1 6  66  876 12636  190296  2935656   45927216
7:  1 7  91 1435 24703  445627  8259727  155635459
8:  1 8 120 2192 43856  922048 19964736  440311936
9:  1 9 153 3177 72441 1739529 43098777 1089331497
...
The triangle a(K,N) = w(N,K-N+1) starts:
K\N 1    2      3      4     5     6    7   8 9 10 ...
0:  1
1:  1    1
2:  1    2      1
3:  1    6      3      1
4:  1   20     15      4     1
5:  1   68     87     28     5     1
6:  1  232    531    232    45     6    1
7:  1  792   3303   2056   485    66    7   1
8:  1 2704  20691  18784  5645   876   91   8 1
9:  1 9232 129951 174112 68245 12636 1435 120 9  1
...
For the graph L_4, shown in the W. Lang link as Figure 3, the counting for round trips of length L=2 for each of the four vertices V_i, i=1..4, reads, from left to right, as follows.
V_1: 1+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)),
this sums to 112, hence the average number is w(4,2)= 112/4 = 28 = a(5,4).

Eric W. Weisstein, from MathWorld: Laguerre Polynomial.
Wolfdieter Lang, Counting walks on Jacobi graphs: an application of orthogonal polynomials.

A201199 (closed Laguerre graphs).

a(K,N) = w(N,K-N+1), with w(N,L) the total number of round trips of length L on the Laguerre graph L_N divided by N (average length L round trip numbers).
The definition of the graph L_N is given as a comment above.
The o.g.f. of w(N,L) is G(N,x) = (1/N)*y*(d/dx)La_N(x)/La_N(x)) with y=1/x. This can be written as
  G(N,x)= 1 + N*La_{N-1}(1/x)/La_N(1/x), where La_N(x) are the monic Laguerre polynomials (see a comment above).

A216294 Triangular array read by rows: T(n,k) is the number of partial permutations of {1,2,...,n} that have exactly k cycles, 0<=k<=n.

Original entry on oeis.org

1, 1, 1, 3, 3, 1, 13, 14, 6, 1, 73, 84, 41, 10, 1, 501, 609, 325, 95, 15, 1, 4051, 5155, 2944, 965, 190, 21, 1, 37633, 49790, 30023, 10689, 2415, 343, 28, 1, 394353, 539616, 340402, 129220, 32179, 5348, 574, 36, 1, 4596553, 6478521, 4246842, 1698374, 455511, 84567, 10794, 906, 45, 1

Offset: 0

Geoffrey Critzer, Sep 04 2012

A partial permutation on a set X is a bijection between two subsets of X.
Row sums are A002720.
First column (corresponding to k=0) is A000262.

			1;
1,     1;
3,     3,   1;
13,   14,   6,  1;
73,   84,  41, 10,  1;
501, 609, 325, 95, 15,  1;

Alois P. Heinz, Rows n = 0..50, flattened
Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics, Cambridge Univ. Press, 2009, page 132.

Cf. A021009. A000262, A002720, A052852, A062147, A062192, A062266, A088699.

gf := exp(x / (1 - x)) / (1 - x)^y:
serx := series(gf, x, 10): poly := n -> simplify(coeff(serx, x, n)):
seq(print(seq(n!*coeff(poly(n), y, k), k = 0..n)), n = 0..9); # Peter Luschny, Feb 23 2023

nn=10;t=Sum[n^(n-1)x^n/n!,{n,1,nn}];f[list_]:=Select[list,#>0&];Map[f,Range[0,nn]!CoefficientList[Series[Exp[ x/(1-x)]/(1-x)^y,{x,0,nn}],{x,y}]]//Flatten

E.g.f.: exp(x/(1-x))/(1-x)^y.
From Peter Bala, Aug 23 2013: (Start)
Exponential Riordan array [exp(x/(1-x)), log(1/(1-x))].
The row polynomials R(n,y), n > = 0, satisfy the 2nd order recurrence equation R(n,y) = (2*n + y - 1)*R(n-1,y) - (n - 1)*(n + y - 2)*R(n-2,y) with R(0,y) = 1 and R(1,y) = 1 + y.
Modulo variations in offset we have: R(n,0) = A000262, R(n,1) = A002720, R(n,2) = A000262, R(n,3) = A052852, R(n,4) = A062147, R(n,5) = A062266 and R(n,6) = A062192. In general, for fixed k, the sequence {R(n,k)}n>=1 gives the entries on a diagonal of the square array A088699. (End)

A286724 Triangle read by rows. A generalization of unsigned Lah numbers, called L[2,1].

Original entry on oeis.org

1, 2, 1, 8, 8, 1, 48, 72, 18, 1, 384, 768, 288, 32, 1, 3840, 9600, 4800, 800, 50, 1, 46080, 138240, 86400, 19200, 1800, 72, 1, 645120, 2257920, 1693440, 470400, 58800, 3528, 98, 1, 10321920, 41287680, 36126720, 12042240, 1881600, 150528, 6272, 128, 1, 185794560, 836075520, 836075520, 325140480, 60963840, 6096384, 338688, 10368, 162, 1, 3715891200, 18579456000, 20901888000, 9289728000, 2032128000, 243855360, 16934400, 691200, 16200, 200, 1

Offset: 0

Wolfdieter Lang, Jun 16 2017

These generalized unsigned Lah numbers are the instance L[2,1] of the Sheffer triangles called L[d,a], with integers d >= 1 and integers 0 <= a < d with gcd(d,a) = 1. The standard unsigned Lah numbers are L[1,0] = A271703.
The Sheffer structure of L[d,a] is ((1 - d*t)^(-2*a/d), t/(1 - d*t)). This follows from the defining property
  risefac[d,a](x, n) = Sum_{m=0..n} L[d,a](n, m)*fallfac[d,a](x, m), where risefac[d,a](x, n):= Product_{0..n-1} (x  + (a+d*j)) for n >= 1 and risefac[d,a](x, 0) := 1, and fallfac[d,a](x, n):= Product_{0..n-1} (x - (a+d*j)) = for n >= 1 and fallfac[d,a](x, 0) := 1. Such rising and falling factorials arise in the generalization of Stirling numbers of both kinds S2[d,a] and S1[d,a]. See the Peter Bala link under A143395 for these falling factorials called there [t;a,b,c]_n with t=x, a=d, b=0, c=a.
In matrix notation: L[d,a] = S1phat[d,a].S2hat[d,a] with the unsigned scaled Stirling1 and the scaled Stirling2 generalizations with Sheffer structures S1phat[d,a] = ((1 - d*t)^(-a/d), -(1/d)*(log(1 - d*t))) and S2hat[d,a] = (exp(a*t), (1/d)*(exp(d*t) - 1). See, e.g., S1phat[2,1] = A028338 and S2hat[2,1] = A039755.
The a- and z-sequences for these Sheffer matrices have e.g.f.s 1 + d*t and ((1 + d*t)/t)*(1 - (1 + d*t)^(-2*a/d)), respectively. See a W. Lang link under A006232 for these types of sequences.
E.g.f. of row polynomials R[d,a](n, x) := Sum_{m=0..n} L[d,a](n, m)*x^m
  (1 - d*x)^(-2*a/d)*exp(t*x/(1 - d*x)) (this is the e.g.f. for the triangle).
E.g.f. of column m: (1 - d*t)^(-2*a/d)*(t/(1 - d*t))^m/m, m >= 0.
Meixner type identity for (monic) row polynomials: (D_x/(1 + d*D_x)) * R[d,a](n, x) = n*R[d,a](n-1, x), n >= 1, with R[d,a](0, x) = 1. The series in the differentiations D_x = d/dx terminates.
General Sheffer recurrence for row polynomials (see the Roman reference, p. 50, Corollary 3.7.2, rewritten for the present Sheffer notation):
  R[d,a](n, x) = [(2*a+x)*1 + 2*d*(a + x)*D_x + d^2*x*(D_x)^2]*R[d,a](n-1, x), n >= 1, with R[d,a](0, x) = 1.
The inverse matrix L^(-1)[d,a] is Sheffer (g[d,a](-t), -f[d,a](-t)) with L[d,a] Sheffer (g[d,a](t), f[d,a](t)) from above. This means (see the column e.g.f. of Sheffer matrices) that L^(-1)[d,a](n, m) = (-1)^(n-m)*L[d,a](n, m). Therefore, the recurrence relations can easily be rewritten for L^(-1)[d,a] by replacing a -> -a and d -> -d.
fallfac[d,a](x, n) = Sum_{m=0..n} L^(-1)[d,a](n, m)*risefac[d,a](x, m), n >= 0.
From Wolfdieter Lang, Aug 12 2017: (Start)
The Sheffer row polynomials R[d,a](n, x) belong to the Boas-Buck class and satisfy therefore the Boas-Buck identity (see the reference, and we use the notation of Rainville, Theorem 50, p. 141, adapted to an exponential generating function) (E_x - n*1)*R[d,a](n, x) = - n*(2*a*1 + d*E_x) * Sum_{k=0..n-1} d^k*R(d,a;n-1-k,x)/(n-1-k)!, with E_x = x*d/dx (Euler operator).
This implies a recurrence for the sequence of column m: L[d,a](n, m) = (n!*(2*a + d*m)/(n-m))*Sum_{p=0..n-1-m} d^p*L[d,a](n-1-p, m)/(n-1-p)!, for n > m>=0, and input L[d,a](m, m) = 1. For the present [d,a] = [2,1] instance see the formula and example sections. (End)
From Wolfdieter Lang, Sep 14 2017: (Start)
The diagonal sequences are 2^D*D!*(binomial(m+D, m))^2, m >= 0, for D >= 0 (main diagonal D = 0). From the o.g.f.s obtained via Lagrange's theorem. See the second W. Lang link below for the general Sheffer case.
The o.g.f. of the diagonal D sequence is 2^D*D!*Sum_{m=0..D} A008459(D, m)*x^m /(1- x)^(2*D + 1),  D >= 0. (End)
It appears that this is also the matrix square of unsigned triangle of coefficients of Laguerre polynomials n!*L_n(x), abs(A021009(n, k)). - Ali Pourzand, Mar 10 2025 [This observation is correct. - Peter Luschny, Mar 10 2025]

			The triangle T(n, m) begins:
  n\m        0         1         2         3        4       5      6     7   8 9
  0:         1
  1:         2         1
  2:         8         8         1
  3:        48        72        18         1
  4:       384       768       288        32        1
  5:      3840      9600      4800       800       50       1
  6:     46080    138240     86400     19200     1800      72      1
  7:    645120   2257920   1693440    470400    58800    3528     98     1
  8:  10321920  41287680  36126720  12042240  1881600  150528   6272   128   1
  9: 185794560 836075520 836075520 325140480 60963840 6096384 338688 10368 162 1
  ...
From _Wolfdieter Lang_, Aug 12 2017: (Start)
Recurrence for column elements with m >= 1, and input column m = 0: T(3, 2) = (3/2)*T(2, 1) + 2*3*T(2, 2) = (3/2)*8 + 6 = 18.
Four term recurrence: T(3, 2) = T(2, 1) + 2*5*T(2, 2) - 4*2^2*T(1, 2) = 8 + 10 + 0 = 18.
Meixner type identity, n=2: 2*R(1, x) = (D_x - 2*(D_x)^2)*R(2, x), 2*(2 + x) = (8 + 2*x) - 2*2.
Sheffer recurrence: R(2, x) = (2 + x)*(2 + x) + 4*(1 + x)*1 + 0 = 8 + 8*x + x^2.
Boas-Buck recurrence for column m = 2 and n = 4: T(4, 2) = (2*4!*3/2)*(1*T(3, 2)/3! + 2*T(2, 2)/2!) = 4!*3*(18/3! + 1) = 288. (End)
Diagonal sequence D = 1: o.g.f. 2*1!*(1 + 1*x)/(1- x)^3 generating
{2*(binomial(m+1, m))^2}_{m >= 0} = {2, 8, 18, 32, ...}. - _Wolfdieter Lang_, Sep 14 2017

Ralph P. Boas, jr. and R. Creighton Buck, Polynomial Expansions of analytic functions, Springer, 1958, pp. 17 - 21, (last sign in eq. (6.11) should be -).
Earl D. Rainville, Special Functions, The Macmillan Company, New York, 1960, ch. 8, sect. 76, 140 - 146.
Steven Roman, The Umbral Calculus, Academic press, Orlando, London, 1984, p. 50.

Peter Bala, The white diamond product of power series
Wolfdieter Lang, On Sums of Powers of Arithmetic Progressions, and Generalized Stirling, Eulerian and Bernoulli Numbers, arXiv:math/1707.04451 [math.NT], July 2017.
Wolfdieter Lang, On Generating functions of Diagonal Sequences of Sheffer and Riordan Number Triangles, arXiv:1708.01421 [math.NT], August 2017.
Emanuele Munarini, Combinatorial identities involving the central coefficients of a Sheffer matrix, Applicable Analysis and Discrete Mathematics (2019) Vol. 13, 495-517.

Cf. A006232, A025167, A028338, A039755, A143395, A271703, A021009.
Column sequences (no leading zeros): A000165, A014479, A286725.
Diagonal sequences: A000012, 2*A000290(m+1), 8*A000537(n+1), 48*A001249, 384*A288876. - Wolfdieter Lang, Sep 14 2017
Row sums are A025167. - Michael Somos, Sep 27 2017

T := (n, k) -> ifelse(n < k, 0, ifelse(k = 0, n!*2^n, (n/k)*T(n-1, k-1) + 2*n*T(n-1, k))): seq(seq(T(n, k), k = 0..n), n = 0..10);  # Peter Luschny, Mar 10 2025

T[ n_, k_] := Coefficient[ Integrate[ Exp[-x^2 - y x] HermiteH[n, x]^2, {x, -Infinity, Infinity}] / (Sqrt[Pi] Exp[y^2 / 4]), y, 2 k]; (* Michael Somos, Sep 27 2017 *)

# Using the function A021009_triangle, displays as a matrix. Following the observation of Ali Pourzand.
print(A021009_triangle(9)^2)  # Peter Luschny, Mar 10 2025

T(n, m) = L[2,1](n, m) = Sum_{k=m..n} A028338(n, k)*A039755(k, m).
Three term recurrence for column elements with m >= 1: T(n, m) = (n/m)*T(n-1, m-1) + 2*n*T(n-1, m) with T(n, m) = 0 for n < m and the column m = 0 is T(n, 0) = (2*n)!! = n*2^n = A000165(n). (From the a- and z-sequences {1, 2, repeat(0)} and {2, repeat(0)}, respectively.)
Four term recurrence: T(n, m) = T(n-1, m-1) + 2*(2*n-1)*T(n-1, m) - 4*(n-1)^2*T(n-2, m), n >= m >= 0, with T(0, 0) = 1, T(-1, m) = 0, T(n, -1) = 0 and T(n, m) = 0 if n < m.
E.g.f. of row polynomials R(n, x) = R[2,1](n, x) (i.e., e.g.f. of the triangle): (1/(1-2*t))*exp(x*t/(1-2*t)).
E.g.f. of column m sequences: (t^m/(1-2*t)^(m+1))/m!, m >= 0.
Meixner type identity: Sum_{k=0..n-1} (-1)^k*2^k*(D_x)^(k+1)*R(n, x) = n*R(n-1, x), n >= 1, with R(0, x) = 1 and D_x = d/dx.
Sheffer recurrence: R(n, x) = [(2 + x)*1 + 4*(1 + x)*D_x + 4*x*(D_x)^2]*R(n-1, x), n >= 1, and R(0, x) = 1.
Boas-Buck recurrence for column m (see a comment above): T(n, m) = (2*n!*(1 + m)/(n-1))*Sum_{p=0..n-1-m} 2^p*T(n-1-p, m)/(n-1-p)!, for n > m >= 0, and input T(m, m) = 1. - Wolfdieter Lang, Aug 12 2017
Explicit form (from the diagonal sequences with the o.g.f.s given as a comment above): T(n, m) = 2^(n-m)*(n-m)!*(binomial(n, n-m))^2 for n >= m >= 0. - Wolfdieter Lang, Sep 23 2017
Let R(n,x) denote the n-th row polynomial. Then x^n*R(n,x) = x^n o x^n, where o denotes the deformed Hadamard product of power series defined in Bala, Section 3.1. - Peter Bala, Jan 18 2018

A295383 a(n) = (2n)! [x^(2n)] (-x/(1 - x))^n/((1 - x)n!).

Original entry on oeis.org

1, -4, 72, -2400, 117600, -7620480, 614718720, -59364264960, 6678479808000, -857813628672000, 123868287980236800, -19863969090648883200, 3502679882984419737600, -673592285189311488000000, 140299650258002307072000000, -31464534897861317399347200000

Offset: 0

Ilya Gutkovskiy, Nov 21 2017

sign

G. C. Greubel, Table of n, a(n) for n = 0..300
Eric Weisstein's World of Mathematics, Laguerre Polynomial
Wikipedia, Laguerre polynomials
Index entries for sequences related to Laguerre polynomials

Central terms of triangles A021009 and A021010.

Cf. A144084.

R:= RealField(); [Round((-16)^n*Gamma(n+1/2)^2/(Pi(R)*Gamma(n+1) )): n in [0..30]]; // G. C. Greubel, Feb 06 2018

a := n -> (-16)^n*GAMMA(n+1/2)^2/(Pi*GAMMA(n+1)):
seq(a(n), n=0..15); # Peter Luschny, Nov 21 2017

Table[(2 n)! SeriesCoefficient[(-x/(1 - x))^n /((1 - x) n!), {x, 0, 2 n}], {n, 0, 15}]
nmax = 15; CoefficientList[Series[2 EllipticK[-16 x]/Pi, {x, 0, nmax}], x] Range[0, nmax]!
Table[(-16)^n*Gamma[n + 1/2]^2/(Pi*Gamma[n + 1]), {n,0,50}] (* G. C. Greubel, Feb 06 2018 *)

for(n=0,30, print1(round((-16)^n*gamma(n+1/2)^2/(Pi*gamma(n+1))), ", ")) \\ G. C. Greubel, Feb 06 2018

E.g.f.: 2*K(-16*x)/Pi, where K() is the complete elliptic integral of the first kind.
a(n) ~ (-1)^n * 16^n * (n-1)! / Pi. - Vaclav Kotesovec, Nov 21 2017
From Peter Luschny, Nov 21 2017: (Start)
a(n) = (-16)^n*Gamma(n+1/2)^2/(Pi*Gamma(n+1)).
a(n) = (-16)^n*binomial(n-1/2,-1/2)*Gamma(n+1/2)/sqrt(Pi).
a(n) ~ (-exp(-1)*n*16)^n/sqrt(n*Pi/2). (End)
a(n) = (-1)^n*binomial(2*n,n)^2*n!. - Alois P. Heinz, Oct 02 2021

A216831 a(n) = Sum_{k=0..n} binomial(n,k)^3 * k!.

Original entry on oeis.org

1, 2, 11, 88, 905, 11246, 162607, 2668436, 48830273, 983353690, 21570885011, 511212091952, 13001401709881, 352856328962918, 10170853073795975, 310093415465876716, 9964607161173899777, 336439048405066012466, 11902368222382731461083, 440122520333417057761160

Offset: 0

Vaclav Kotesovec, Sep 17 2012

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A000522, A002720, A000172, A119400, A206178, A021009.

Table[Sum[Binomial[n, k]^3*k!, {k, 0, n}], {n, 0, 25}]
Table[HypergeometricPFQ[{-n, -n, -n}, {1}, -1], {n, 0, 20}] (* Vladimir Reshetnikov, Sep 28 2016 *)

a(n) = sum(k=0, n, binomial(n,k)^3 * k!); \\ Michel Marcus, May 04 2021

Recurrence: (8*n^2+31*n+21)*a(n+3) - (24*n^3+157*n^2+308*n+162)*a(n+2) + (24*n^4+117*n^3+178*n^2+71*n-18)*a(n+1) - (8*n^2+31*n+30)*(n+1)^3*a(n) = 0.
a(n) ~ n^(n-1/6)/(sqrt(6*Pi)*exp(n+n^(1/3)-3*n^(2/3)-1/3)). - Vaclav Kotesovec, Sep 30 2012
a(n) = hypergeom([-n, -n, -n], [1], -1). - Vladimir Reshetnikov, Sep 28 2016
a(n) = Sum_{k=0..n} binomial(n, k)*|A021009(n, k)|. - Peter Luschny, May 04 2021
Sum_{n>=0} a(n) * x^n / n!^3 = BesselI(0,2*sqrt(x)) * Sum_{n>=0} x^n / n!^3. - Ilya Gutkovskiy, Jun 19 2022

A222469 Denominator sequence of the n-th convergent of the continued fraction 1/(1 - 2/(2 - 2/(3 - 2/(4 - ...)))).

Original entry on oeis.org

1, 1, 0, -2, -8, -36, -200, -1328, -10224, -89360, -873152, -9425952, -111365120, -1428894656, -19781794944, -293869134848, -4662342567680, -78672085380864, -1406772851720192, -26571340011921920, -528613254534998016

Offset: 0

Gary Detlefs and Wolfdieter Lang, Mar 23 2013

The corresponding numerator sequence is A222470(n).
a(n) = Q(n,-2) with the denominator polynomials Q of A084950. All the given formulas follow from there. The limit of the continued fraction (-1/2)*(0 + K_{k=1..oo} (-2/k)) = 1/(1 - 2/(2 - 2/(3 - 2/(4 - ...)))) is (+1/2)*sqrt(2)*BesselJ(1,2*sqrt(2))/BesselJ(0,2*sqrt(2)) = -1.43974932187... For more decimals see A222471.
For a combinatorial interpretation in terms of labeled Morse codes see a comment on A084950. Here each dash has label x = -2, and the dots have label j if they are at position j. Labels are multiplied and for a(n) all labeled codes on [1,2,...,n] have to be summed.

			a(4) = 4*a(3) - 2*a(2) = 4*(-2) + 2*0 = -8.
Continued fraction convergent: 1/(1 - 2/(2 - 2/(3 - 2/4))) = -3/2 = -12/8 = A222470(4)/a(4).
Morse code: a(4) = -8 from the sum of all 5 labeled codes on [1,2,3,4], one with no dash, three with one dash and one with two dashes: 4! + (3*4 + 1*4 + 1*2)*(-2) + (-2)^2 = -8.

G. C. Greubel, Table of n, a(n) for n = 0..445

Cf. A084950, A221913, A222470, A222471.

Cf. A001040(n+1) (x=1), A058797 (x=-1), A222467 (x=2).

I:=[1, 1]; [n le 2 select I[n] else n*Self(n-1) - 2*Self(n-2): n in [1..30]]; // G. C. Greubel, May 17 2018

RecurrenceTable[{a[0] == 1, a[1] == 1, a[n] == n*a[n - 1] - 2 a[n - 2]}, a[n], {n, 50}] (* G. C. Greubel, Aug 16 2017 *)

m=30; v=concat([1,1], vector(m-2)); for(n=3, m, v[n]=n*v[n-1] -2*v[n-2]); v \\ G. C. Greubel, May 17 2018

a(n) = n*a(n-1) - 2*a(n-2), a(-1) = 0, a(0) = 1, n >= 1.
a(n) = Sum_{m=0..floor(n/2)} a(n-m, m)*(-2)^m, n >= 0, with a(n,m) = (n!/m!)*binomial(n,m) = |A021009(n,m)| (Laguerre).
a(n) = Pi*(z/2)^(n+1)*(BesselY(0,z)*BesselJ(n+1,z) - BesselJ(0,z)*BesselY(n+1,z)) with z := 2*sqrt(2).
E.g.f.: Pi*c/(2*sqrt(1-z))*(BesselJ(1, c*sqrt(1-z))*BesselY(0, c) - BesselY(1, c*sqrt(1-z))*BesselJ(0, c)), with c = 2*sqrt(2).
Asymptotics: lim_{n->oo} a(n)/n! = BesselJ(0, 2*sqrt(2)) = -0.1965480950...

A001811 Coefficients of Laguerre polynomials.

Original entry on oeis.org

1, 25, 450, 7350, 117600, 1905120, 31752000, 548856000, 9879408000, 185513328000, 3636061228800, 74373979680000, 1586644899840000, 35272336619520000, 816302647480320000, 19645683716026368000, 491142092900659200000, 12740803704070041600000

Offset: 4

N. J. A. Sloane

			G.f. = x^4 + 25*x^5 + 450*x^6 + 7350*x^7 + 117600*x^8 + 1905120*x^9 + ...

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 799.
Cornelius Lanczos, Applied Analysis. Prentice-Hall, Englewood Cliffs, NJ, 1956, p. 519.
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).

T. D. Noe, Table of n, a(n) for n = 4..100
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Cornelius Lanczos, Applied Analysis. (Annotated scans of selected pages)
Index entries for sequences related to Laguerre polynomials.

Cf. A053495.

Cf. A001113, A001620, A091725, A099285.

with(combstruct):ZL:=[st, {st=Prod(left, right), left=Set(U, card=r+2), right=Set(U, card=1)}, labeled]: subs(r=2, stack): seq(count(subs(r=2, ZL), size=m), m=4..19) ; # Zerinvary Lajos, Feb 07 2008

Table[n! n (n - 1) (n - 2) (n - 3)/(4!)^2, {n, 4, 20}] (* T. D. Noe, Aug 10 2012 *)

[factorial(m) * binomial(m, 4) / 24 for m in range(4,19)] # Zerinvary Lajos, Jul 05 2008

a(n) = n!*n*(n-1)(n-2)(n-3)/(4!)^2. a(4)=1, a(n+1) = a(n) * (n+1)^2 / (n-3).
a(n) = A021009(n, 4), n >= 4.
E.g.f.: x^4/(4!*(1-x)^5).
If we define f(n,i,x) = Sum_{k=i..n} Sum_{j=i..k} binomial(k,j) * Stirling1(n,k) * Stirling2(j,i) * x^(k-j) then a(n) = (-1)^n*f(n,4,-5), (n >= 4). - Milan Janjic, Mar 01 2009
From Amiram Eldar, May 02 2022: (Start)
Sum_{n>=4} 1/a(n) = 64*(Ei(1) - gamma - e) + 272/3, where Ei(1) = A091725, gamma = A001620, and e = A001113.
Sum_{n>=4} (-1)^n/a(n) = 544*(gamma - Ei(-1)) - 320/e - 944/3, where Ei(-1) = -A099285. (End)

More terms from Larry Reeves (larryr(AT)acm.org), Feb 07 2001

Corrected by T. D. Noe, Aug 10 2012

A025166 E.g.f.: -exp(-x/(1-2x))/(1-2x).

Original entry on oeis.org

-1, -1, -1, 7, 127, 1711, 23231, 334391, 5144063, 84149983, 1446872959, 25661798119, 454494403199, 7489030040207, 89680375568447, -759618144120809, -127049044802971649, -7480338932613448769, -369274690558092738817, -17262533154073740329017

Offset: 0

Wouter Meeussen

sign

Polynomials in A021009 evaluated at 2.

Vaclav Kotesovec, Table of n, a(n) for n = 0..400

Cf. A025167, A025168.

a := n -> -(-2)^n*KummerU(-n, 1, 1/2):
seq(simplify(a(n)), n=0..19); # Peter Luschny, Feb 12 2020

Table[ -n! 2^n LaguerreL[ n, 1/2 ], {n, 0, 12} ]

Conjecture: a(n) + (-4*n+3)*a(n-1) + 4*(n-1)^2*a(n-2) = 0. - R. J. Mathar, Feb 05 2013
a(n) = -(-2)^n*KummerU(-n, 1, 1/2). - Peter Luschny, Feb 12 2020
Sum_{n>=0} a(n) * x^n / (n!)^2 = -exp(2*x) * BesselJ(0,2*sqrt(x)). - Ilya Gutkovskiy, Jul 17 2020

Corrected and extended by Vladeta Jovovic, Jan 29 2003

A199577 Coefficient triangle of the associated Laguerre polynomials of order 1.

Original entry on oeis.org

1, -3, 1, 11, -8, 1, -50, 58, -15, 1, 274, -444, 177, -24, 1, -1764, 3708, -2016, 416, -35, 1, 13068, -33984, 23544, -6560, 835, -48, 1, -109584, 341136, -288360, 101560, -17370, 1506, -63, 1, 1026576, -3733920, 3736440, -1595040, 343410, -39900, 2513, -80, 1

Offset: 0

Wolfdieter Lang, Nov 25 2011

Associated monic polynomials of order m (a nonnegative integer) in one variable, p_n(m;x), have the coefficients of the three-term recurrence of the original orthogonal monic polynomials p_n(x)=p_n(0;x) shifted by +m, and use the same inputs for n=-1 and n=0, namely 0 and 1, respectively. See, e.g., the Ismail reference, p. 27, Definition (2.3.4), where the notation is P_n(x;c) = p_n(c;x).
p_n(x)=p_n(0;x) and p_{n-1}(1;x) provide the fundamental system for the three-term recurrence of p_n(x) with general input.
p_{n-1}(1;x)/p_n(0;x) is the n-th approximation to the Jacobi continued fraction related to the three-term recurrence.
The monic row polynomials are La_n(1;x) = Sum_{k=0..n} a(n,k)*x^k, with the monic Laguerre polynomials La_n(x), which have the three-term recurrence
  La_n(x) = (x - (2*n - 1))*La_{n-1}(x) - (n - 1)^2*La_{n-2}(x), La_{-1}(x) = 0, La_0(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;x)= L_n^{(alpha=0)}(x;1)*(n+1)!*(-1)^n.
From Wolfdieter Lang, Dec 04 2011: (Start)
The e.g.f. gLa(z,x) for La_n(1;x) can be obtained from the o.g.f. G(z,x) for the non-monic version L_n^{(alpha=0)}(x;1) by gLa(z,x)=(d/dz)(z*G(-z,x)).
  G(z,x) satisfies the ordinary first-order inhomogeneous differential equation, derived from the recurrence:
  (d/dz)G(z,x) = (2/(1-z)+(1-x)/(1-z)^2-1/(z*(1-z)^2))* G(z,x) + 1/(z*(1-z)^2), with G(0,x)=1. The standard solution is:
  G(z,x) = exp(-x/(1-z))*(Ei(1,-x) - Ei(1,-x/(1-z)))/(z*(1-z)), with the exponential integral Ei(1,y) = Integral_{t = y..oo} exp(-t)/t. From this the e.g.f. gLa(z,x), given in the Formula section, results. (End)
Note that these are not the associated Laguerre polynomials as usually defined in the literature. - Peter Bala, Mar 22 2024

			n\k      0        1       2        3      4    5  6   7
0:       1
1:      -3        1
2:      11       -8       1
3:     -50       58     -15        1
4:     274     -444     177      -24      1
5:   -1764     3708   -2016      416    -35    1
6:   13068   -33984   23544    -6560    835  -48   1
7: -109584   341136 -288360   101560 -17370 1506 -63  1
...

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

Cf. A021009 (Laguerre), A199578 (row sums), A002793(n+1)*(-1)^n (alternating row sums, conjecture). [This conjecture has been proved by Wolfdieter Lang, Dec 12 2011]

Ei[1, y_] := -ExpIntegralEi[-y];
(cc = CoefficientList[#, x]; cc (Length[cc]-1)!)& /@ CoefficientList[(1 - Exp[-x/(1+z)] (1 - x/(1+z)) (Ei[1, -x/(1+z)] - Ei[1, -x]))/(1+z)^2 + O[z]^9, z] // Flatten (* Jean-François Alcover, Oct 28 2019 *)

p(n) = numerator(bestapprPade(Ser(vector(2*n, k, (k-1)!))));
concat(vector(9, n, Vec((-1)^(n-1)*p(n))))  \\ Gheorghe Coserea, Dec 01 2016

T(n,k) = [x^k] La_n(1; x), n >= 0, k = 0,...,n, with the three-term recurrence of the row polynomials: La_n(1; x) = (x - (2*n + 1))*La_{n-1}(1; x) - (n^2)* La_{n-2}(1; x), with La_{-1}(1; x) = 0, La_0(1; x) = 1.
The e.g.f. for La_n(1; x) is (1 - exp(-x/(1+z))*(1-x/(1+z))*(Ei(1,-x/(1+z)) - Ei(1,-x)))/(1+z)^2, with the exponential integral Ei. See the comments section for the definition and the proof. - Wolfdieter Lang, Dec 04 2011
n-th row polynomial = (-1)^n*(n+1)!*L(n+1, x)*Sum_{k = 0..n+1} 1/(k*L(k, x)*L(k-1, x)), where L(k, x) denotes the k-th Laguerre polynomial. See A021009. - Peter Bala, Mar 22 2024

A001812 Coefficients of Laguerre polynomials.

Original entry on oeis.org

1, 36, 882, 18816, 381024, 7620480, 153679680, 3161410560, 66784798080, 1454424491520, 32724551059200, 761589551923200, 18341615042150400, 457129482588979200, 11787410229615820800, 314330939456421888000, 8663746518767628288000, 246661959710796005376000

Offset: 5

N. J. A. Sloane

nonn

			G.f. = x^5 + 36*x^6 + 882*x^7 + 18816*x^8 + 381024*x^9 + 7620480*x^10 + ...

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 799.
Cornelius Lanczos, Applied Analysis. Prentice-Hall, Englewood Cliffs, NJ, 1956, p. 519.
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).

T. D. Noe, Table of n, a(n) for n = 5..100
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Cornelius Lanczos, Applied Analysis (Annotated scans of selected pages)
Index entries for sequences related to Laguerre polynomials.

Cf. A001113, A001620, A021009, A091725, A099285.

[((Factorial(n)/Factorial(5))^2)/Factorial(n-5): n in [5..20]]; // G. C. Greubel, May 11 2018

Table[((n!/5!)^2)/(n-5)!, {n, 5, 20}] (* T. D. Noe, Aug 10 2012 *)

for(n=5,20, print1(((n!/5!)^2)/(n-5)!, ", ")) \\ G. C. Greubel, May 11 2018

[factorial(m) * binomial(m, 5) / 120 for m in range(5,23)] # Zerinvary Lajos, Jul 05 2008

a(n) = (-1)*A021009(n, 5), n >= 5.
a(n) = ((n!/5!)^2)/(n-5)!, n >= 5.
If we define f(n,i,x) = Sum_{k=i..n} Sum_{j=i..k} binomial(k,j)*Stirling1(n,k)*Stirling2(j,i)*x^(k-j) then a(n) = (-1)^(n-1)*f(n,5,-6), (n>=5). - Milan Janjic, Mar 01 2009
From Amiram Eldar, May 02 2022: (Start)
Sum_{n>=5} 1/a(n) = 375*(gamma - Ei(1)) + 150*e + 175/2, where e = A001113, gamma = A001620, and Ei(1) = A091725.
Sum_{n>=5} (-1)^(n+1)/a(n) = 5225*(gamma - Ei(-1)) - 3100/e - 18125/6, where Ei(-1) = -A099285. (End)

cp's OEIS Frontend

A201198 Triangle version of the array w(N,L) of the average number of round trips of length L on Laguerre graphs L_N.

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A216294 Triangular array read by rows: T(n,k) is the number of partial permutations of {1,2,...,n} that have exactly k cycles, 0<=k<=n.

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A286724 Triangle read by rows. A generalization of unsigned Lah numbers, called L[2,1].

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A295383 a(n) = (2*n)! * [x^(2*n)] (-x/(1 - x))^n/((1 - x)*n!).

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A216831 a(n) = Sum_{k=0..n} binomial(n,k)^3 * k!.

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A222469 Denominator sequence of the n-th convergent of the continued fraction 1/(1 - 2/(2 - 2/(3 - 2/(4 - ...)))).

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A001811 Coefficients of Laguerre polynomials.

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A295383 a(n) = (2n)! [x^(2n)] (-x/(1 - x))^n/((1 - x)n!).

A025166 E.g.f.: -exp(-x/(1-2x))/(1-2x).