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A154556 Exponential Riordan array [exp(-x), x(1+x/2)].

%I A154556 #9 Jul 22 2025 06:18:25
%S A154556 1,-1,1,1,-1,1,-1,0,0,1,1,2,-3,2,1,-1,-5,5,-5,5,1,1,9,0,-5,0,9,1,-1,
%T A154556 -14,-21,35,-35,21,14,1,1,20,70,-56,35,-56,70,20,1,-1,-27,-162,-42,
%U A154556 189,-189,42,162,27,1
%N A154556 Exponential Riordan array [exp(-x), x(1+x/2)].
%C A154556 Row sums are double factorials A001147 (aerated).
%C A154556 Product A007318*(this triangle) is A122848.
%F A154556 T(n,k) = (n!/k!)*sum{j=0..k, (-1)^(n-k)*C(k,j)*(-1/2)^j/(n-k-j)!};
%F A154556 T(n,k) = sum{j=0..n, (-1)^(n-j)*C(n,j)*C(j,k)*k!/((2k-j)!*2^(j-k))};
%e A154556 Triangle begins
%e A154556 1,
%e A154556 -1, 1,
%e A154556 1, -1, 1,
%e A154556 -1, 0, 0, 1,
%e A154556 1, 2, -3, 2, 1,
%e A154556 -1, -5, 5, -5, 5, 1,
%e A154556 1, 9, 0, -5, 0, 9, 1,
%e A154556 -1, -14, -21, 35, -35, 21, 14, 1,
%e A154556 1, 20, 70, -56, 35, -56, 70, 20, 1,
%e A154556 -1, -27, -162, -42, 189, -189, 42, 162, 27, 1,
%e A154556 1, 35, 315, 510, -735, 693, -735, 510, 315, 35, 1
%t A154556 (* The function RiordanArray is defined in A256893. *)
%t A154556 RiordanArray[E^-#&, # (1 + #/2)&, 10, True] // Flatten (* _Jean-François Alcover_, Jul 19 2019 *)
%Y A154556 Cf. A001147, A007318, A122848.
%K A154556 easy,sign,tabl
%O A154556 0,12
%A A154556 _Paul Barry_, Jan 11 2009