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A104558 Triangle, read by rows, equal to the matrix inverse of A104557 and related to Laguerre polynomials.

%I A104558 #11 Sep 08 2022 08:45:17
%S A104558 1,-1,1,0,-2,1,0,2,-4,1,0,0,6,-6,1,0,0,-6,18,-9,1,0,0,0,-24,36,-12,1,
%T A104558 0,0,0,24,-96,72,-16,1,0,0,0,0,120,-240,120,-20,1,0,0,0,0,-120,600,
%U A104558 -600,200,-25,1,0,0,0,0,0,-720,1800,-1200,300,-30,1,0,0,0,0,0,720,-4320,5400,-2400,450,-36,1,0,0,0,0,0,0,5040,-15120,12600,-4200,630,-42,1
%N A104558 Triangle, read by rows, equal to the matrix inverse of A104557 and related to Laguerre polynomials.
%C A104558 Even-indexed rows are found in A066667 (generalized Laguerre polynomials). Odd-indexed rows are found in A021009 (Laguerre polynomials L_n(x)). Row sums equal A056920 (offset 1). Absolute row sums equal A056953 (offset 1).
%H A104558 G. C. Greubel, <a href="/A104558/b104558.txt">Rows n=0..100 of triangle, flattened</a>
%F A104558 T(n, k) = (-1)^(n-k)*(n-k)!*C(1+[n/2], k+1-[(n+1)/2])*C([(n+1)/2], k-[n/2]).
%e A104558 Rows begin:
%e A104558   1;
%e A104558   -1,1;
%e A104558   0,-2,1;
%e A104558   0,2,-4,1;
%e A104558   0,0,6,-6,1;
%e A104558   0,0,-6,18,-9,1;
%e A104558   0,0,0,-24,36,-12,1;
%e A104558   0,0,0,24,-96,72,-16,1;
%e A104558   0,0,0,0,120,-240,120,-20,1;
%e A104558   0,0,0,0,-120,600,-600,200,-25,1;
%e A104558   ...
%e A104558 Unsigned columns read downwards equals rows of matrix inverse A104557 read backwards:
%e A104558   1;
%e A104558   1,1;
%e A104558   2,2,1;
%e A104558   6,6,4,1;
%e A104558   24,24,18,6,1;
%e A104558   120,120,96,36,9,1;
%e A104558   ...
%t A104558 T[n_, k_] := (-1)^(n - k)*(n - k)!*Binomial[1 + Floor[n/2], k + 1 - Floor[(n + 1)/2]]*Binomial[Floor[(n+1)/2], k -Floor[n/2]]; Table[T[n, k], {n, 0, 20}, {k, 0, n}]//Flatten (* _G. C. Greubel_, May 14 2018 *)
%o A104558 (PARI) {T(n,k)=(-1)^(n-k)*(n-k)!*binomial(1+n\2,k+1-(n+1)\2)* binomial( (n+1)\2,k-n\2)};
%o A104558 (Magma) /* As triangle */ [[(-1)^(n-k)*Factorial(n-k)*Binomial(1+ Floor(n/2), k +1 -Floor((n+1)/2))*Binomial(Floor((n+1)/2), k - Floor(n/2)): k in [0..n]]: n in [0..10]]; // _G. C. Greubel_, May 14 2018
%Y A104558 Cf. A104557, A066667, A021009, A056920, A056953.
%K A104558 sign,tabl
%O A104558 0,5
%A A104558 _Paul D. Hanna_, Mar 16 2005