A123202 Triangle of coefficients of n!*(1 - x)^n*L_n(x/(1 - x)), where L_n(x) is the Laguerre polynomial.

1, 1, -2, 2, -8, 7, 6, -36, 63, -34, 24, -192, 504, -544, 209, 120, -1200, 4200, -6800, 5225, -1546, 720, -8640, 37800, -81600, 94050, -55656, 13327, 5040, -70560, 370440, -999600, 1536150, -1363572, 653023, -130922, 40320, -645120, 3951360, -12794880

Offset: 0

Author

Roger L. Bagula, Oct 04 2006

Keywords

• sign
• tabl

Comments

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).

Examples

			Triangle begins:
       1;
       1,    -2;
       2,    -8,     7;
       6,   -36,    63,    -34;
      24,  -192,   504,   -544,   209;
     120, -1200,  4200,  -6800,  5225,  -1546;
     720, -8640, 37800, -81600, 94050, -55656, 13327;
      ... reformatted. - _Franck Maminirina Ramaharo_, Oct 13 2018
		

References

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

Links

• G. C. Greubel, Table of n, a(n) for n = 0..5150 (Rows n=0..100 of triangle, flattened; offset corrected by _Georg Fischer_, Jan 31 2019)
• 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].
• Eric Weisstein's World of Mathematics, Laguerre Polynomial

Crossrefs

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

Programs

Maple

M := (n,x) -> n!*subs(x=(x/(1-x)),orthopoly[L](n,x))*(1-x)^n:
seq(print(seq(coeff(simplify(M(n,x)),x,k),k=0..n)),n=0..6); # Peter Luschny, Jan 05 2015

Mathematica

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

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 */

PARI

row(n) = Vecrev(n!*(1-x)^n*pollaguerre(n, 0, x/(1 - x))); \\ Michel Marcus, Feb 06 2021

Formula

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

From Franck Maminirina Ramaharo, Oct 13 2018: (Start)

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

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

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

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

Extensions

Edited by N. J. A. Sloane, Jun 12 2007

Edited, new name, and offset corrected by Franck Maminirina Ramaharo, Oct 13 2018