A123202 - OEIS

%I #42 Feb 06 2021 08:07:52

%S 1,1,-2,2,-8,7,6,-36,63,-34,24,-192,504,-544,209,120,-1200,4200,-6800,

%T 5225,-1546,720,-8640,37800,-81600,94050,-55656,13327,5040,-70560,

%U 370440,-999600,1536150,-1363572,653023,-130922,40320,-645120,3951360,-12794880

%N Triangle of coefficients of n!*(1 - x)^n*L_n(x/(1 - x)), where L_n(x) is the Laguerre polynomial.

%C The n-th row consists of the coefficients in the expansion of Sum_{j=0..n} A021009(n,j)*x^j*(1 - x)^(n - j).

%D Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, 9th printing. New York: Dover, 1972, p. 782.

%D Gengzhe Chang and Thomas W. Sederberg, Over and Over Again, The Mathematical Association of America, 1997, p. 164, figure 26.1.

%H G. C. Greubel, <a href="/A123202/b123202.txt">Table of n, a(n) for n = 0..5150</a> (Rows n=0..100 of triangle, flattened; offset corrected by _Georg Fischer_, Jan 31 2019)

%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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LaguerrePolynomial.html">Laguerre Polynomial</a>

%F T(n, k) = [x^k] (n!*L_n(x)*(1 - x)^n) with L_n(x) the Laguerre polynomial after substituting x by x/(1 - x). - _Peter Luschny_, Jan 05 2015

%F From _Franck Maminirina Ramaharo_, Oct 13 2018: (Start)

%F G.f.: exp(-x*y/(1 - (1 - x)*y))/(1 - (1 - x)*y).

%F T(n,1) = A000142(n).

%F T(n,2) = -A052582(n).

%F T(n,n) = A002720(n). (End)

%e Triangle begins:

%e        1;

%e        1,    -2;

%e        2,    -8,     7;

%e        6,   -36,    63,    -34;

%e       24,  -192,   504,   -544,   209;

%e      120, -1200,  4200,  -6800,  5225,  -1546;

%e      720, -8640, 37800, -81600, 94050, -55656, 13327;

%e       ... reformatted. - _Franck Maminirina Ramaharo_, Oct 13 2018

%p M := (n,x) -> n!*subs(x=(x/(1-x)),orthopoly[L](n,x))*(1-x)^n:

%p seq(print(seq(coeff(simplify(M(n,x)),x,k),k=0..n)),n=0..6); # _Peter Luschny_, Jan 05 2015

%t w = Table[n!*CoefficientList[LaguerreL[n, x], x], {n, 0, 10}];

%t v = Table[CoefficientList[Sum[w[[n + 1]][[m + 1]]*x^ m*(1 - x)^(n - m), {m, 0, n}], x], {n, 0, 10}]; Flatten[v]

%o (Maxima) create_list(ratcoef(n!*(1 - x)^n*laguerre(n, x/(1 - x)), x, k), n, 0, 10, k, 0, n); /* _Franck Maminirina Ramaharo_, Oct 13 2018 */

%o (PARI) row(n) = Vecrev(n!*(1-x)^n*pollaguerre(n, 0, x/(1 - x))); \\ _Michel Marcus_, Feb 06 2021

%Y Cf. A122753, A123018, A123019, A123021, A123027, A123199, A123202, A123217, A123221.

%K sign,tabl

%O 0,3

%A _Roger L. Bagula_, Oct 04 2006

%E Edited by _N. J. A. Sloane_, Jun 12 2007

%E Edited, new name, and offset corrected by _Franck Maminirina Ramaharo_, Oct 13 2018