A066667 Coefficient triangle of generalized Laguerre polynomials (a=1).

1, 2, -1, 6, -6, 1, 24, -36, 12, -1, 120, -240, 120, -20, 1, 720, -1800, 1200, -300, 30, -1, 5040, -15120, 12600, -4200, 630, -42, 1, 40320, -141120, 141120, -58800, 11760, -1176, 56, -1, 362880, -1451520, 1693440, -846720, 211680, -28224, 2016

Offset: 0

Author

Christian G. Bower, Dec 17 2001

Keywords

• sign
• tabl

Comments

Same as A008297 (Lah triangle) except for signs.

Row sums give A066668. Unsigned row sums give A000262.

The Laguerre polynomials L(n;x;a=1) under discussion are connected with Hermite-Bell polynomials p(n;x) for power -1 (see also A215216) defined by the following relation: p(n;x) := x^(2n)*exp(x^(-1))*(d^n exp(-x^(-1))/dx^n). We have L(n;x;a=1)=(-x)^(n-1)*p(n;1/x), p(n+1;x)=x^2(dp(n;x)/dx)+(1-2*n*x)p(n;x), and p(1;x)=1, p(2;x)=1-2*x, p(3;x)=1-6*x+6*x^2, p(4;x)=1-12*x+36*x^2-24*x^3, p(5;x)=1-20*x+120*x^2-240*x^3+120*x^4. Note that if we set w(n;x):=x^(2n)*p(n;1/x) then w(n+1;x)=(w(n;x)-(dw(n;x)/dx))*x^2, which is almost analogous to the recurrence formula for Bell polynomials B(n+1;x)=(B(n;x)+(dB(n;x)/dx))*x. - Roman Witula, Aug 06 2012.

Examples

			Triangle a(n,m) begins
n\m     0        1       2       3      4      5    6   7  8
0:      1
1:      2       -1
2:      6       -6       1
3:     24      -36      12      -1
4:    120     -240     120     -20      1
5:    720    -1800    1200    -300     30     -1
6:   5040   -15120   12600   -4200    630    -42    1
7:  40320  -141120  141120  -58800  11760  -1176   56  -1
8: 362880 -1451520 1693440 -846720 211680 -28224 2016 -72  1
9: 3628800, -16329600, 21772800, -12700800, 3810240, -635040, 60480, -3240, 90, -1.
Reformatted and extended by _Wolfdieter Lang_, Jan 31 2013.
From _Wolfdieter Lang_, Jan 31 2013 (Start)
Recurrence (standard): a(4,2) = 2*4*12 - (-36) - 4*3*1 = 120.
Recurrence (simple): a(4,2) = 7*12 - (-36) = 120. (End)
		

References

• M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 778 (22.5.17).
• F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Cambridge, 1998, p. 95 (4.1.62)
• R. Witula, E. Hetmaniok, and D. Slota, The Hermite-Bell polynomials for negative powers, (submitted, 2012)

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 >= n >= 150, flattened).
• 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].
• M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 778 (22.5.17).
• Mathias Pétréolle and Alan D. Sokal, Lattice paths and branched continued fractions. II. Multivariate Lah polynomials and Lah symmetric functions, arXiv:1907.02645 [math.CO], 2019.
• Jian Zhou, On Some Mathematics Related to the Interpolating Statistics, arXiv:2108.10514 [math-ph], 2021.

Crossrefs

Cf. A021009, A062137, A062138, A062139, A062140, A089231.

Programs

Maple

A066667 := (n, k) -> (-1)^k*binomial(n, k)*(n + 1)!/(k + 1)!:
for n from 0 to 9 do seq(A066667(n,k), k = 0..n) od; # Peter Luschny, Jun 22 2022

Mathematica

Table[(-1)^m*Binomial[n, m]*(n + 1)!/(m + 1)!, {n, 0, 8}, {m, 0, n}] // Flatten (* Michael De Vlieger, Sep 04 2019 *)

PARI

row(n) = Vecrev(n!*pollaguerre(n, 1)); \\ Michel Marcus, Feb 06 2021

Formula

E.g.f. (relative to x, keep y fixed): A(x)=(1/(1-x))^2*exp(x*y/(x-1)).

From Wolfdieter Lang, Jan 31 2013: (Start)

a(n,m) = (-1)^m*binomial(n,m)*(n+1)!/(m+1)!, n >= m >= 0. [corrected by Georg Fischer, Oct 26 2022]

Recurrence from standard three term recurrence for orthogonal generalized Laguerre polynomials {P(n,x):=n!*L(1,n,x)}:

P(n,x) = (2*n-x)*P(n-1,x) - n*(n-1)*P(n-2), n>=1, P(-1,x) = 0, P(0,x) = 1.

a(n,m) = 2*n*a(n-1,m) - a(n-1,m-1) - n*(n-1)*a(n-2,m), n>=1, a(0,0) =1, a(n,-1) = 0, a(n,m) = 0 if n < m.

Simplified recurrence from explicit form of a(n,m):

a(n,m) = (n+m+1)*a(n-1,m) - a(n-1,m-1), n >= 1, a(0,0) =1, a(n,-1) = 0, a(n,m) = 0 if n < m.

(End)