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

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

Offset

0,5

Comments

Inverse matrix of A099567. Row sums are A099570.

Links

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Formula

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

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

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

From G. C. Greubel, Jul 25 2022: (Start)

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

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

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

Example

Rows begin as:
    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,   15,  -6,   1;
  -16,  27,  -36,  41,  -36,  21,  -7,  1;
   22, -43,   63, -77,   77, -57,  28, -8,  1;
  -29,  65, -106, 140, -154, 134, -85, 36, -9,  1;

Maple

C := proc (n, k) if 0 <= k and k <= n then factorial(n)/(factorial(k)*factorial(n-k)) else 0 end if;
end proc:
for n from 0 to 10 do
    seq((-1)^(n+k)*(C(n, n-k) + add((i-2)*C(n-i, n-k-i), i = 3..n)), k = 0..n);
end do; # Peter Bala, Mar 21 2018

Mathematica

T[n_, k_]:= (-1)^(n+k)*(Binomial[n, k] + Binomial[n-1, k+2]);
Table[T[n, k], {n, 0, 15}, {k, 0, n}]//Flatten (* G. C. Greubel, Jul 25 2022 *)

Prog

(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
(SageMath)
def A099569(n, k): return 1 if (n==0) else (-1)^(n+k)*(binomial(n, k) +binomial(n-1, k+2))
flatten([[A099569(n, k) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Jul 25 2022

Crossrefs

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

Sequence in context: A202191 A052250 A333878 * A191579 A097724 A091836

Adjacent sequences: A099566 A099567 A099568 * A099570 A099571 A099572

Keyword

easy,sign,tabl

Author

Paul Barry, Oct 22 2004

Status

approved