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A123202
Triangle of coefficients of n!*(1 - x)^n*L_n(x/(1 - x)), where L_n(x) is the Laguerre polynomial.
13
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
0,3
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).
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
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)
EXAMPLE
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
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]
PROG
(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
KEYWORD
sign,tabl
AUTHOR
Roger L. Bagula, Oct 04 2006
EXTENSIONS
Edited by N. J. A. Sloane, Jun 12 2007
Edited, new name, and offset corrected by Franck Maminirina Ramaharo, Oct 13 2018
STATUS
approved