A021012 Triangle of coefficients in expansion of x^n in terms of Laguerre polynomials L_n(x).

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• sign
• tabl
• easy
• nice

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

Examples

			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).
		

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

Crossrefs

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

Cf. A000165 (row sum of absolute values).

Cf. A136572.

Programs

Magma

[[(-1)^k*Factorial(n)*Binomial(n,k): k in [0..n]]: n in [0..10]]; // G. C. Greubel, Feb 06 2018

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 *)

PARI

for(n=0,10, for(k=0,n, print1((-1)^k*n!*binomial(n,k), ", "))) \\ G. C. Greubel, Feb 06 2018

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

Extensions

More terms from Vladeta Jovovic, May 11 2003