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A021012
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Triangle of coefficients in expansion of x^n in terms of Laguerre polynomials L_n(x).
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9
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1, 1, -1, 2, -4, 2, 6, -18, 18, -6, 24, -96, 144, -96, 24, 120, -600, 1200, -1200, 600, -120, 720, -4320, 10800, -14400, 10800, -4320, 720, 5040, -35280, 105840, -176400, 176400, -105840, 35280, -5040, 40320, -322560, 1128960, -2257920, 2822400, -2257920, 1128960, -322560, 40320, 362880, -3265920
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OFFSET
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0,4
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COMMENTS
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Triangle T(n,k), read by rows: given by [1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, ...] DELTA [ -1, -1, -2, -2, -3, -3, -4, -4, -5, -5, ...], where DELTA is the operator defined in A084938. - Philippe Deléham, Feb 14 2005
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REFERENCES
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M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 799.
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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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EXAMPLE
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Triangle begins:
1;
1, -1;
2, -4, 2;
6, -18, 18, -6;
24, -96, 144, -96, 24;
...
x^3 = 6*LaguerreL(0,x) - 18*LaguerreL(1,x) + 18*LaguerreL(2,x) - 6*LaguerreL(3,x).
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MATHEMATICA
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row[n_] := Table[ a[n, k], {k, 0, n}] /. SolveAlways[ x^n == Sum[ a[n, k]*LaguerreL[k, x], {k, 0, n}], x] // First; (* or, after Vladeta Jovovic: *) row[n_] := Table[(-1)^k*n!*Binomial[n, k], {k, 0, n}]; Table[ row[n], {n, 0, 9}] // Flatten (* Jean-François Alcover, Oct 05 2012 *)
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PROG
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(PARI) for(n=0, 10, for(k=0, n, print1((-1)^k*n!*binomial(n, k), ", "))) \\ G. C. Greubel, Feb 06 2018
(Magma) [[(-1)^k*Factorial(n)*Binomial(n, k): k in [0..n]]: n in [0..10]]; // G. C. Greubel, Feb 06 2018
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CROSSREFS
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Cf. A000165 (row sum of absolute values).
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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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