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 A021012 Triangle of coefficients in expansion of x^n in terms of Laguerre polynomials L_n(x). 9
 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 A361727 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 August 6 06:59 EDT 2024. Contains 374960 sequences. (Running on oeis4.)