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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 (list; table; graph; refs; listen; history; text; internal format)
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

CROSSREFS

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

Sequence in context: A245582 A197820 A064862 * A195299 A095297 A269545

Adjacent sequences:  A123199 A123200 A123201 * A123203 A123204 A123205

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

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Last modified May 15 12:37 EDT 2021. Contains 343920 sequences. (Running on oeis4.)