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Coefficient triangle of generalized Laguerre polynomials (a=1).
21

%I #48 Oct 26 2022 12:03:10

%S 1,2,-1,6,-6,1,24,-36,12,-1,120,-240,120,-20,1,720,-1800,1200,-300,30,

%T -1,5040,-15120,12600,-4200,630,-42,1,40320,-141120,141120,-58800,

%U 11760,-1176,56,-1,362880,-1451520,1693440,-846720,211680,-28224,2016

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

%C Same as A008297 (Lah triangle) except for signs.

%C Row sums give A066668. Unsigned row sums give A000262.

%C 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 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 F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Cambridge, 1998, p. 95 (4.1.62)

%D R. Witula, E. Hetmaniok, and D. Slota, The Hermite-Bell polynomials for negative powers, (submitted, 2012)

%H Michael De Vlieger, <a href="/A066667/b066667.txt">Table of n, a(n) for n = 0..11475</a> (rows 0 >= n >= 150, flattened).

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

%F From _Wolfdieter Lang_, Jan 31 2013: (Start)

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

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

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

%F 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 (End)

%e Triangle a(n,m) begins

%e n\m 0 1 2 3 4 5 6 7 8

%e 0: 1

%e 1: 2 -1

%e 2: 6 -6 1

%e 3: 24 -36 12 -1

%e 4: 120 -240 120 -20 1

%e 5: 720 -1800 1200 -300 30 -1

%e 6: 5040 -15120 12600 -4200 630 -42 1

%e 7: 40320 -141120 141120 -58800 11760 -1176 56 -1

%e 8: 362880 -1451520 1693440 -846720 211680 -28224 2016 -72 1

%e 9: 3628800, -16329600, 21772800, -12700800, 3810240, -635040, 60480, -3240, 90, -1.

%e Reformatted and extended by _Wolfdieter Lang_, Jan 31 2013.

%e From _Wolfdieter Lang_, Jan 31 2013 (Start)

%e Recurrence (standard): a(4,2) = 2*4*12 - (-36) - 4*3*1 = 120.

%e Recurrence (simple): a(4,2) = 7*12 - (-36) = 120. (End)

%p A066667 := (n, k) -> (-1)^k*binomial(n, k)*(n + 1)!/(k + 1)!:

%p for n from 0 to 9 do seq(A066667(n,k), k = 0..n) od; # _Peter Luschny_, Jun 22 2022

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

%o (PARI) row(n) = Vecrev(n!*pollaguerre(n, 1)); \\ _Michel Marcus_, Feb 06 2021

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

%K sign,tabl

%O 0,2

%A _Christian G. Bower_, Dec 17 2001