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A343861 Coefficient triangle of generalized Laguerre polynomials n!*L(n,n,x) (rising powers of x).

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