OFFSET
0,2
COMMENTS
Same as A008297 (Lah triangle) except for signs.
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.
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.
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)
EXAMPLE
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)
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 *)
PROG
(PARI) row(n) = Vecrev(n!*pollaguerre(n, 1)); \\ Michel Marcus, Feb 06 2021
CROSSREFS
KEYWORD
sign,tabl
AUTHOR
Christian G. Bower, Dec 17 2001
STATUS
approved