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A021012 Triangle of coefficients in expansion of x^n in terms of Laguerre polynomials L_n(x). 7
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

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

REFERENCES

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.

LINKS

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

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].

Index entries for sequences related to Laguerre polynomials

FORMULA

T(n, k) = (-1)^k*n!*binomial(n, k). - Vladeta Jovovic, May 11 2003

Sum_{k>=0} T(n, k)*T(m, k) = (n+m)!. - Philippe Deléham, Feb 14 2005

Unsigned sequence = A136572 * A007318 - Gary W. Adamson, Jan 07 2008

A136572*PS, where PS is a triangle with PS[n,k] = (-1)^k*A007318[n,k]. PS = 1/PS. - Gerald McGarvey, Aug 20 2009

EXAMPLE

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).

MATHEMATICA

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 *)

PROG

(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

CROSSREFS

Columns include (essentially) A000142, A001563, A001804, A001805, A001806, A001807.

Cf. A000165 (row sum of absolute values).

Cf. A136572.

Sequence in context: A253666 A174298 A196347 * A229460 A154120 A261964

Adjacent sequences:  A021009 A021010 A021011 * A021013 A021014 A021015

KEYWORD

sign,tabl,easy,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Vladeta Jovovic, May 11 2003

STATUS

approved

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Last modified May 31 13:01 EDT 2020. Contains 334748 sequences. (Running on oeis4.)