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A100218 Riordan array ((1-2*x)/(1-x), (1-x)).

%I A100218 #28 Apr 28 2024 11:34:03
%S A100218 1,-1,1,-1,-2,1,-1,0,-3,1,-1,0,2,-4,1,-1,0,0,5,-5,1,-1,0,0,-2,9,-6,1,
%T A100218 -1,0,0,0,-7,14,-7,1,-1,0,0,0,2,-16,20,-8,1,-1,0,0,0,0,9,-30,27,-9,1,
%U A100218 -1,0,0,0,0,-2,25,-50,35,-10,1,-1,0,0,0,0,0,-11,55,-77,44,-11,1,-1,0,0,0,0,0,2,-36,105,-112,54,-12,1
%N A100218 Riordan array ((1-2*x)/(1-x), (1-x)).
%H A100218 G. C. Greubel, <a href="/A100218/b100218.txt">Rows n = 0..49 of the triangle, flattened, flattened</a>
%H A100218 P. Peart and W.-J. Woan, <a href="http://dx.doi.org/10.1016/0166-218X(99)00166-3">A divisibility property for a subgroup of Riordan matrices</a>, Discrete Applied Mathematics, Vol. 98, Issue 3, Jan 2000, 255-263.
%F A100218 Sum_{k=0..n} T(n, k) = A100219(n) (row sums).
%F A100218 Number triangle T(n, k) = (-1)^(n-k)*(binomial(k, n-k) + binomial(k-1, n-k-1)), with T(0, 0) = 1. - _Paul Barry_, Nov 09 2004
%F A100218 T(n,k) = T(n-1,k) + T(n-1,k-1) - 2*T(n-2,k-1) + T(n-3,k-1), T(0,0)=1, T(1,0)=-1, T(1,1)=1, T(n,k)=0 if k < 0 or if k > n. - _Philippe Deléham_, Jan 09 2014
%F A100218 From _G. C. Greubel_, Mar 28 2024: (Start)
%F A100218 T(n, n-1) = A000027(n), n >= 1.
%F A100218 T(n, n-2) = -A080956(n-1), n >= 2.
%F A100218 T(2*n, n) = A280560(n).
%F A100218 T(2*n-1, n) = A157142(n-1), n >= 1.
%F A100218 T(2*n+1, n) = -A000007(n) = A154955(n+2).
%F A100218 T(3*n, n) = T(4*n, n) = A000007(n).
%F A100218 Sum_{k=0..n} (-1)^k*T(n, k) = A355021(n).
%F A100218 Sum_{k=0..floor(n/2)} T(n-k, k) = (-1)^n*A098601(n).
%F A100218 Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = -1 + 2*A077961(n) + A077961(n-2). (End)
%F A100218 From _Peter Bala_, Apr 28 2024: (Start)
%F A100218 This Riordan array has the form ( x*h'(x)/h(x), h(x) ) with h(x) = x*(1 - x) and hence belongs to the hitting time subgroup of the Riordan group (see Peart and Woan for properties of this subgroup).
%F A100218 T(n,k) = [x^(n-k)] (1/c(x))^n, where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108. In general the (n, k)-th entry of the hitting time array ( x*h'(x)/h(x), h(x) ) has the form [x^(n-k)] f(x)^n, where f(x) = x/( series reversion of h(x) ). (End)
%e A100218 Triangle begins as:
%e A100218    1;
%e A100218   -1,  1;
%e A100218   -1, -2,  1;
%e A100218   -1,  0, -3,  1;
%e A100218   -1,  0,  2, -4,  1;
%e A100218   -1,  0,  0,  5, -5,   1;
%e A100218   -1,  0,  0, -2,  9,  -6,   1;
%e A100218   -1,  0,  0,  0, -7,  14,  -7,  1;
%e A100218   -1,  0,  0,  0,  2, -16,  20, -8,  1;
%e A100218   -1,  0,  0,  0,  0,   9, -30, 27, -9,  1;
%t A100218 T[0,0]:= 1; T[1,1]:= 1; T[1,0]:= -1; T[n_, k_]:= T[n, k]= If[k<0 || k>n, 0, T[n- 1,k] +T[n-1,k-1] -2*T[n-2,k-1] +T[n-3,k-1]]; Table[T[n,k], {n,0,14}, {k,0,n} ]//Flatten (* _G. C. Greubel_, Mar 13 2017 *)
%o A100218 (Magma)
%o A100218 A100218:= func< n,k | n eq 0 select 1 else (-1)^(n+k)*(Binomial(k,n-k) + Binomial(k-1,n-k-1)) >;
%o A100218 [A100218(n,k): k in [0..n], n in [0..13]]; // _G. C. Greubel_, Mar 28 2024
%o A100218 (SageMath)
%o A100218 def A100218(n,k): return 1 if n==0 else (-1)^(n+k)*(binomial(k,n-k) + binomial(k-1,n-k-1))
%o A100218 flatten([[A100218(n,k) for k in range(n+1)] for n in range(14)]) # _G. C. Greubel_, Mar 28 2024
%Y A100218 Cf. A000007, A000027, A077961, A080956, A098601.
%Y A100218 Cf. A154955, A157142, A280560, A355021.
%Y A100218 Row sums are A100219.
%Y A100218 Matrix inverse of A100100.
%Y A100218 Apart from signs, same as A098599.
%Y A100218 Very similar to triangle A111125.
%K A100218 easy,sign,tabl
%O A100218 0,5
%A A100218 _Paul Barry_, Nov 08 2004