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A099569 Riordan array (((1+x)^2 - x^3)/(1+x)^3, 1/(1+x)).

%I A099569 #12 Dec 31 2023 16:16:42
%S A099569 1,-1,1,1,-2,1,-2,3,-3,1,4,-5,6,-4,1,-7,9,-11,10,-5,1,11,-16,20,-21,
%T A099569 15,-6,1,-16,27,-36,41,-36,21,-7,1,22,-43,63,-77,77,-57,28,-8,1,-29,
%U A099569 65,-106,140,-154,134,-85,36,-9,1,37,-94,171,-246,294,-288,219,-121,45,-10,1,-46,131,-265,417,-540,582,-507,340,-166,55,-11,1
%N A099569 Riordan array (((1+x)^2 - x^3)/(1+x)^3, 1/(1+x)).
%C A099569 Inverse matrix of A099567. Row sums are A099570.
%H A099569 G. C. Greubel, <a href="/A099569/b099569.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A099569 Sum_{k=0..n} T(n, k) = A099570(n).
%F A099569 Columns have g.f. ((1+x)^2 - x^3)/(1+x)^3*(x/(1+x))^k.
%F A099569 T(n,k) = (-1)^(n+k)*(binomial(n, n-k) + Sum_{i = 3..n} (i-2)*binomial(n-i,n-k-i)), for 0 <= k <= n, otherwise 0. - _Peter Bala_, Mar 21 2018
%F A099569 From _G. C. Greubel_, Jul 25 2022: (Start)
%F A099569 T(n, k) = (-1)^(n+k)*(binomial(n, k) + binomial(n-1, k+2)), with T(0, k) = 1.
%F A099569 T(2*n-1, n-1) = (-1)^n*A076540(n), n >= 1.
%F A099569 T(n, n-1) = -n. (End)
%e A099569 Rows begin as:
%e A099569     1;
%e A099569    -1,   1;
%e A099569     1,  -2,    1;
%e A099569    -2,   3,   -3,   1;
%e A099569     4,  -5,    6,  -4,    1;
%e A099569    -7,   9,  -11,  10,   -5,   1;
%e A099569    11, -16,   20, -21,   15,  -6,   1;
%e A099569   -16,  27,  -36,  41,  -36,  21,  -7,  1;
%e A099569    22, -43,   63, -77,   77, -57,  28, -8,  1;
%e A099569   -29,  65, -106, 140, -154, 134, -85, 36, -9,  1;
%p A099569 C := proc (n, k) if 0 <= k and k <= n then factorial(n)/(factorial(k)*factorial(n-k)) else 0 end if;
%p A099569 end proc:
%p A099569 for n from 0 to 10 do
%p A099569     seq((-1)^(n+k)*(C(n, n-k) + add((i-2)*C(n-i, n-k-i), i = 3..n)), k = 0..n);
%p A099569 end do; # _Peter Bala_, Mar 21 2018
%t A099569 T[n_, k_]:= (-1)^(n+k)*(Binomial[n, k] + Binomial[n-1, k+2]);
%t A099569 Table[T[n, k], {n,0,15}, {k,0,n}]//Flatten (* _G. C. Greubel_, Jul 25 2022 *)
%o A099569 (Magma) [n eq 0 select 1 else (-1)^(n+k)*(Binomial(n, k) + Binomial(n-1, k+2)): k in [0..n], n in [0..15]]; // _G. C. Greubel_, Jul 25 2022
%o A099569 (SageMath)
%o A099569 def A099569(n, k): return 1 if (n==0) else (-1)^(n+k)*(binomial(n, k) +binomial(n-1, k+2))
%o A099569 flatten([[A099569(n,k) for k in (0..n)] for n in (0..15)]) # _G. C. Greubel_, Jul 25 2022
%Y A099569 Cf. A076540, A099570 (row sums), A099567.
%K A099569 easy,sign,tabl
%O A099569 0,5
%A A099569 _Paul Barry_, Oct 22 2004