A177767 Triangle read by rows: T(n,k) = binomial(n - 1, k - 1), 1 <= k <= n, and T(n,0) = A153881(n+1), n >= 0.

1, -1, 1, -1, 1, 1, -1, 1, 2, 1, -1, 1, 3, 3, 1, -1, 1, 4, 6, 4, 1, -1, 1, 5, 10, 10, 5, 1, -1, 1, 6, 15, 20, 15, 6, 1, -1, 1, 7, 21, 35, 35, 21, 7, 1, -1, 1, 8, 28, 56, 70, 56, 28, 8, 1, -1, 1, 9, 36, 84, 126, 126, 84, 36, 9, 1, -1, 1, 10, 45, 120, 210, 252, 210, 120, 45, 10, 1

Offset

0,9

Comments

Row sums yield A000225 preceded by 1.

Except for signs, this is A135225.

Links

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

Formula

Row n = coefficients in the expansion of x*(1 + x)^(n - 1) - 1, n > 0.

From Franck Maminirina Ramaharo, Oct 23 2018: (Start)

G.f.: (1 - 3*y + (2 + x)*y^2)/(1 - (2 + x)*y + (1 + x)*y^2).

E.g.f.: (2 + x - (1 + x)*exp(y) + x*exp((1 + x)*y))/(1 + x). (End)

From G. C. Greubel, Apr 22 2024: (Start)

Sum_{k=0..n} (-1)^k*T(n, k) = A153881(n+1) - [n=1].

Sum_{k=0..floor(n/2)} T(n-k, k) = A000071(n-1) + [n=0].

Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = -A131026(n-1) + [n=0]. (End)

Example

Triangle begins:
   1;
  -1, 1;
  -1, 1, 1;
  -1, 1, 2,  1;
  -1, 1, 3,  3,  1;
  -1, 1, 4,  6,  4,   1;
  -1, 1, 5, 10, 10,   5,   1;
  -1, 1, 6, 15, 20,  15,   6,  1;
  -1, 1, 7, 21, 35,  35,  21,  7,  1;
  -1, 1, 8, 28, 56,  70,  56, 28,  8, 1;
  -1, 1, 9, 36, 84, 126, 126, 84, 36, 9, 1;
   ...

Mathematica

Flatten[Table[If[n == 0, {1}, CoefficientList[x*(1 + x)^( n - 1) - 1, x]], {n, 0, 10}]]

Prog

(Maxima)
T(n, k) := if k = 0 then 2*floor(1/(n + 1)) - 1 else binomial(n - 1, k - 1)$
create_list(T(n, k), n, 0, 12, k, 0, n); /* Franck Maminirina Ramaharo, Oct 23 2018 */
(Magma)
A177767:= func< n, k | k eq n select 1 else  k eq 0 select -1 else Binomial(n-1, k-1) >;
[A177767(n, k): k in [0..n], n in [0..13]]; // G. C. Greubel, Apr 22 2024
(SageMath)
flatten([[binomial(n-1, k-1) - int(k==0) + 2*int(n==0) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Apr 22 2024

Crossrefs

Cf. A000225, A000071, A007318, A014473, A097805, A131026, A135225.

Sequence in context: A297299 A135225 A208891 * A047030 A047120 A096751

Adjacent sequences: A177764 A177765 A177766 * A177768 A177769 A177770

Keyword

sign,tabl,easy

Author

Roger L. Bagula, May 13 2010

Extensions

Edited and new name by Franck Maminirina Ramaharo, Oct 23 2018

Status

approved