|
%I #29 Aug 12 2022 00:33:22
%S 1,2,-1,12,-8,1,120,-90,18,-1,1680,-1344,336,-32,1,30240,-25200,7200,
%T -900,50,-1,665280,-570240,178200,-26400,1980,-72,1,17297280,
%U -15135120,5045040,-840840,76440,-3822,98,-1,518918400,-461260800,161441280,-29352960,3057600,-188160,6720,-128,1
%N Coefficient triangle of generalized Laguerre polynomials n!*L(n,n,x) (rising powers of x).
%H G. C. Greubel, <a href="/A343861/b343861.txt">Rows n = 0..50 of the triangle, flattened</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Laguerre_polynomials">Laguerre polynomials</a>
%H <a href="/index/La#Laguerre">Index entries for sequences related to Laguerre polynomials</a>
%F T(n, k) = (-1)^k * n! * binomial(2*n,n-k)/k! = (-1)^k * (2*n)! * binomial(n,k)/(k+n)!.
%F T(n, 0) = A001813(n).
%F T(n, 1) = -A092956(n-1).
%F Sum_{k=0..n} T(n, k) = A006902(n).
%F Sum_{k=0..n} (-1)^k * T(n, k) = A082545(n).
%e The triangle begins:
%e 1;
%e 2, -1;
%e 12, -8, 1;
%e 120, -90, 18, -1;
%e 1680, -1344, 336, -32, 1;
%e 30240, -25200, 7200, -900, 50, -1;
%e 665280, -570240, 178200, -26400, 1980, -72, 1;
%t T[n_, k_] := (-1)^k * (2*n)! * Binomial[n, k]/(k + n)!; Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* _Amiram Eldar_, May 11 2021 *)
%o (PARI) T(n, k) = (-1)^k*(2*n)!*binomial(n,k)/(k+n)!;
%o (PARI) row(n) = Vecrev(n!*pollaguerre(n, n));
%o (Magma) [(-1)^k*Factorial(n-k)*Binomial(n,k)*Binomial(2*n, n+k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Aug 11 2022
%o (SageMath)
%o def A343861(n,k): return (-1)^k*factorial(n-k)*binomial(n,k)*binomial(2*n,n+k)
%o flatten([[A343861(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Aug 11 2022
%Y For k=0..1 the (unsigned) columns give A001813, A092956(n-1).
%Y Row sums (signed) give A006902, row sums (unsigned) give A082545.
%Y Cf. A066667, A062137, A062138, A062139, A062140.
%K sign,tabl
%O 0,2
%A _Seiichi Manyama_, May 01 2021
|
