A021010 Triangle of coefficients of Laguerre polynomials L_n(x) (powers of x in decreasing order).

1, -1, 1, 1, -4, 2, -1, 9, -18, 6, 1, -16, 72, -96, 24, -1, 25, -200, 600, -600, 120, 1, -36, 450, -2400, 5400, -4320, 720, -1, 49, -882, 7350, -29400, 52920, -35280, 5040, 1, -64, 1568, -18816, 117600, -376320, 564480, -322560, 40320, -1, 81, -2592, 42336, -381024, 1905120, -5080320, 6531840, -3265920, 362880

Offset: 0

Author

N. J. A. Sloane

Keywords

• sign
• tabl
• easy
• nice

Comments

abs(T(n,k)) = k!*binomial(n,k)^2 = number of k-matchings of the complete bipartite graph K_{n,n}. Example: abs(T(2,2))=2 because in the bipartite graph K_{2,2} with vertex sets {A,B},{A',B'} we have the 2-matchings {AA',BB'} and {AB',BA'}. Row sums of the absolute values yield A002720. - Emeric Deutsch, Dec 25 2004

Examples

			   1;
  -1,   1;
   1,  -4,   2;
  -1,   9, -18,   6;
   1, -16,  72, -96,  24;
  ...
		

References

• 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.

Links

• T. D. Noe, Rows n = 0..50 of triangle, 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].
• J. Fernando Barbero G., Jesús Salas, Eduardo J. S. Villaseñor, Bivariate Generating Functions for a Class of Linear Recurrences. I. General Structure, arXiv:1307.2010 [math.CO], 2013.
• C. Lanczos, Applied Analysis (Annotated scans of selected pages) See page 519.
• Eric Weisstein's World of Mathematics, Rook Polynomial
• Kin Yip Wong, A Dynamic Coupling Model of Optical Conductivity in Mixed-Valence Systems, arXiv:2410.13144 [cond-mat.mtrl-sci], 2024. See p. 12.
• Index entries for sequences related to Laguerre polynomials

Crossrefs

Cf. A002720, A021009, A009940 (row sums).

Central terms: A295383.

Programs

Magma

[[(-1)^(n-k)*Factorial(k)*Binomial(n, k)^2: k in [0..n]]: n in [0..10]]; // G. C. Greubel, Feb 06 2018

Maple

T:=(n,k)->(-1)^(n-k)*k!*binomial(n,k)^2: for n from 0 to 9 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form # Emeric Deutsch, Dec 25 2004

Mathematica

Flatten[ Table[ Reverse[ CoefficientList[n!*LaguerreL[n, x], x]], {n, 0, 9}]] (* Jean-François Alcover, Nov 24 2011 *)

PARI

LaguerreL(n,v='x) = {
  my(x='x+O('x^(n+1)), t='t);
  subst(polcoeff(exp(-x*t/(1-x))/(1-x), n), 't, v);
};
concat(apply(n->Vec(n!*LaguerreL(n)), [0..9])) \\ Gheorghe Coserea, Oct 26 2017

PARI

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

Formula

T(n, k) = (-1)^(n-k)*k!*binomial(n, k)^2. - Emeric Deutsch, Dec 25 2004