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

%I #12 Dec 31 2023 16:16:42

%S 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 15,-6,1,-16,27,-36,41,-36,21,-7,1,22,-43,63,-77,77,-57,28,-8,1,-29,

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

%C Inverse matrix of A099567. Row sums are A099570.

%H G. C. Greubel, <a href="/A099569/b099569.txt">Rows n = 0..50 of the triangle, flattened</a>

%F Sum_{k=0..n} T(n, k) = A099570(n).

%F Columns have g.f. ((1+x)^2 - x^3)/(1+x)^3*(x/(1+x))^k.

%F 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 From _G. C. Greubel_, Jul 25 2022: (Start)

%F T(n, k) = (-1)^(n+k)*(binomial(n, k) + binomial(n-1, k+2)), with T(0, k) = 1.

%F T(2*n-1, n-1) = (-1)^n*A076540(n), n >= 1.

%F T(n, n-1) = -n. (End)

%e Rows begin as:

%e 1;

%e -1, 1;

%e 1, -2, 1;

%e -2, 3, -3, 1;

%e 4, -5, 6, -4, 1;

%e -7, 9, -11, 10, -5, 1;

%e 11, -16, 20, -21, 15, -6, 1;

%e -16, 27, -36, 41, -36, 21, -7, 1;

%e 22, -43, 63, -77, 77, -57, 28, -8, 1;

%e -29, 65, -106, 140, -154, 134, -85, 36, -9, 1;

%p C := proc (n, k) if 0 <= k and k <= n then factorial(n)/(factorial(k)*factorial(n-k)) else 0 end if;

%p end proc:

%p for n from 0 to 10 do

%p seq((-1)^(n+k)*(C(n, n-k) + add((i-2)*C(n-i, n-k-i), i = 3..n)), k = 0..n);

%p end do; # _Peter Bala_, Mar 21 2018

%t T[n_, k_]:= (-1)^(n+k)*(Binomial[n, k] + Binomial[n-1, k+2]);

%t Table[T[n, k], {n,0,15}, {k,0,n}]//Flatten (* _G. C. Greubel_, Jul 25 2022 *)

%o (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 (SageMath)

%o def A099569(n, k): return 1 if (n==0) else (-1)^(n+k)*(binomial(n, k) +binomial(n-1, k+2))

%o flatten([[A099569(n,k) for k in (0..n)] for n in (0..15)]) # _G. C. Greubel_, Jul 25 2022

%Y Cf. A076540, A099570 (row sums), A099567.

%K easy,sign,tabl

%O 0,5

%A _Paul Barry_, Oct 22 2004