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%I #37 Aug 14 2022 16:04:39
%S 1,1,-1,2,-4,2,6,-18,18,-6,24,-96,144,-96,24,120,-600,1200,-1200,600,
%T -120,720,-4320,10800,-14400,10800,-4320,720,5040,-35280,105840,
%U -176400,176400,-105840,35280,-5040,40320,-322560,1128960,-2257920,2822400,-2257920,1128960,-322560,40320,362880,-3265920
%N Triangle of coefficients in expansion of x^n in terms of Laguerre polynomials L_n(x).
%C 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
%D 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.
%H G. C. Greubel, <a href="/A021012/b021012.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H <a href="/index/La#Laguerre">Index entries for sequences related to Laguerre polynomials</a>
%F T(n, k) = (-1)^k*n!*binomial(n, k). - _Vladeta Jovovic_, May 11 2003
%F Sum_{k>=0} T(n, k)*T(m, k) = (n+m)!. - _Philippe Deléham_, Feb 14 2005
%F Unsigned sequence = A136572 * A007318 - _Gary W. Adamson_, Jan 07 2008
%F A136572*PS, where PS is a triangle with PS[n,k] = (-1)^k*A007318[n,k]. PS = 1/PS. - _Gerald McGarvey_, Aug 20 2009
%e Triangle begins:
%e 1;
%e 1, -1;
%e 2, -4, 2;
%e 6, -18, 18, -6;
%e 24, -96, 144, -96, 24;
%e ...
%e x^3 = 6*LaguerreL(0,x) - 18*LaguerreL(1,x) + 18*LaguerreL(2,x) - 6*LaguerreL(3,x).
%t 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 *)
%o (PARI) for(n=0,10, for(k=0,n, print1((-1)^k*n!*binomial(n,k), ", "))) \\ _G. C. Greubel_, Feb 06 2018
%o (Magma) [[(-1)^k*Factorial(n)*Binomial(n,k): k in [0..n]]: n in [0..10]]; // _G. C. Greubel_, Feb 06 2018
%Y Columns include (essentially) A000142, A001563, A001804, A001805, A001806, A001807.
%Y Cf. A000165 (row sum of absolute values).
%Y Cf. A136572.
%K sign,tabl,easy,nice
%O 0,4
%A _N. J. A. Sloane_
%E More terms from _Vladeta Jovovic_, May 11 2003
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