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A123202
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Triangle of coefficients of n!*(1 - x)^n*L_n(x/(1 - x)), where L_n(x) is the Laguerre polynomial.
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13
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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
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OFFSET
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0,3
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COMMENTS
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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).
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REFERENCES
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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.
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LINKS
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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].
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FORMULA
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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
G.f.: exp(-x*y/(1 - (1 - x)*y))/(1 - (1 - x)*y).
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EXAMPLE
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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;
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MAPLE
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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
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MATHEMATICA
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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]
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PROG
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(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
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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