cp's OEIS Frontend

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

%I A066667 #48 Oct 26 2022 12:03:10
%S A066667 1,2,-1,6,-6,1,24,-36,12,-1,120,-240,120,-20,1,720,-1800,1200,-300,30,
%T A066667 -1,5040,-15120,12600,-4200,630,-42,1,40320,-141120,141120,-58800,
%U A066667 11760,-1176,56,-1,362880,-1451520,1693440,-846720,211680,-28224,2016
%N A066667 Coefficient triangle of generalized Laguerre polynomials (a=1).
%C A066667 Same as A008297 (Lah triangle) except for signs.
%C A066667 Row sums give A066668. Unsigned row sums give A000262.
%C A066667 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.
%D A066667 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).
%D A066667 F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Cambridge, 1998, p. 95 (4.1.62)
%D A066667 R. Witula, E. Hetmaniok, and D. Slota, The Hermite-Bell polynomials for negative powers, (submitted, 2012)
%H A066667 Michael De Vlieger, <a href="/A066667/b066667.txt">Table of n, a(n) for n = 0..11475</a> (rows 0 >= n >= 150, flattened).
%H A066667 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H A066667 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 778 (22.5.17).
%H A066667 Mathias Pétréolle and Alan D. Sokal, <a href="https://arxiv.org/abs/1907.02645">Lattice paths and branched continued fractions. II. Multivariate Lah polynomials and Lah symmetric functions</a>, arXiv:1907.02645 [math.CO], 2019.
%H A066667 Jian Zhou, <a href="https://arxiv.org/abs/2108.10514">On Some Mathematics Related to the Interpolating Statistics</a>, arXiv:2108.10514 [math-ph], 2021.
%F A066667 E.g.f. (relative to x, keep y fixed): A(x)=(1/(1-x))^2*exp(x*y/(x-1)).
%F A066667 From _Wolfdieter Lang_, Jan 31 2013: (Start)
%F A066667 a(n,m) = (-1)^m*binomial(n,m)*(n+1)!/(m+1)!, n >= m >= 0. [corrected by _Georg Fischer_, Oct 26 2022]
%F A066667 Recurrence from standard three term recurrence for orthogonal generalized Laguerre polynomials {P(n,x):=n!*L(1,n,x)}:
%F A066667   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.
%F A066667   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.
%F A066667 Simplified recurrence from explicit form of a(n,m):
%F A066667   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.
%F A066667 (End)
%e A066667 Triangle a(n,m) begins
%e A066667 n\m     0        1       2       3      4      5    6   7  8
%e A066667 0:      1
%e A066667 1:      2       -1
%e A066667 2:      6       -6       1
%e A066667 3:     24      -36      12      -1
%e A066667 4:    120     -240     120     -20      1
%e A066667 5:    720    -1800    1200    -300     30     -1
%e A066667 6:   5040   -15120   12600   -4200    630    -42    1
%e A066667 7:  40320  -141120  141120  -58800  11760  -1176   56  -1
%e A066667 8: 362880 -1451520 1693440 -846720 211680 -28224 2016 -72  1
%e A066667 9: 3628800, -16329600, 21772800, -12700800, 3810240, -635040, 60480, -3240, 90, -1.
%e A066667 Reformatted and extended by _Wolfdieter Lang_, Jan 31 2013.
%e A066667 From _Wolfdieter Lang_, Jan 31 2013 (Start)
%e A066667 Recurrence (standard): a(4,2) = 2*4*12 - (-36) - 4*3*1 = 120.
%e A066667 Recurrence (simple): a(4,2) = 7*12 - (-36) = 120. (End)
%p A066667 A066667 := (n, k) -> (-1)^k*binomial(n, k)*(n + 1)!/(k + 1)!:
%p A066667 for n from 0 to 9 do seq(A066667(n,k), k = 0..n) od; # _Peter Luschny_, Jun 22 2022
%t A066667 Table[(-1)^m*Binomial[n, m]*(n + 1)!/(m + 1)!, {n, 0, 8}, {m, 0, n}] // Flatten (* _Michael De Vlieger_, Sep 04 2019 *)
%o A066667 (PARI) row(n) = Vecrev(n!*pollaguerre(n, 1)); \\ _Michel Marcus_, Feb 06 2021
%Y A066667 Cf. A021009, A062137, A062138, A062139, A062140, A089231.
%K A066667 sign,tabl
%O A066667 0,2
%A A066667 _Christian G. Bower_, Dec 17 2001