A021010 - OEIS

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

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OFFSET

0,5

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

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

FORMULA

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

EXAMPLE

-1, 1;

1, -4, 2;

-1, 9, -18, 6;

1, -16, 72, -96, 24;

...

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 *)

PROG

(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

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

CROSSREFS