A110555 Triangle of partial sums of alternating binomial coefficients: T(n, k) = Sum_{j=0..k} binomial(n, j)*(-1)^j, for n >= 0, 0 <= k <= n.

1, 1, 0, 1, -1, 0, 1, -2, 1, 0, 1, -3, 3, -1, 0, 1, -4, 6, -4, 1, 0, 1, -5, 10, -10, 5, -1, 0, 1, -6, 15, -20, 15, -6, 1, 0, 1, -7, 21, -35, 35, -21, 7, -1, 0, 1, -8, 28, -56, 70, -56, 28, -8, 1, 0, 1, -9, 36, -84, 126, -126, 84, -36, 9, -1, 0, 1, -10, 45, -120, 210, -252, 210, -120

Offset: 0

Author

Reinhard Zumkeller, Jul 27 2005

Keywords

• sign
• easy
• tabl

Examples

			Triangle T(n, k) starts:
  [0] 1;
  [1] 1,  0;
  [2] 1, -1,  0;
  [3] 1, -2,  1,   0;
  [4] 1, -3,  3,  -1,  0;
  [5] 1, -4,  6,  -4,  1,   0;
  [6] 1, -5, 10, -10,  5,  -1,  0;
  [7] 1, -6, 15, -20, 15,  -6,  1,  0;
  [8] 1, -7, 21, -35, 35, -21,  7, -1,  0.
		

Links

• G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
• Ângela Mestre and José Agapito, A Family of Riordan Group Automorphisms, J. Int. Seq., Vol. 22 (2019), Article 19.8.5.
• Index entries for triangles and arrays related to Pascal's triangle

Crossrefs

T(n,1) = -n + 1 for n>0;

T(n,2) = A000217(n-2) for n > 1;

T(n,3) = -A000292(n-4) for n > 2;

T(n,4) = A000332(n-1) for n > 3;

T(n,5) = -A000389(n-1) for n > 5;

T(n,6) = A000579(n-1) for n > 6;

T(n,7) = -A000580(n-1) for n > 7;

T(n,8) = A000581(n-1) for n > 8;

T(n,9) = -A000582(n-1) for n > 9;

T(n,10) = A001287(n-1) for n > 10;

T(n,11) = -A001288(n-1) for n > 11;

T(n,12) = A010965(n-1) for n > 12;

T(n,13) = -A010966(n-1) for n > 13;

T(n,14) = A010967(n-1) for n > 14;

T(n,15) = -A010968(n-1) for n > 15;

T(n,16) = A010969(n-1) for n > 16.

Cf. A071919 (variant), A000007 (row sums), A110556 (central terms).

Cf. A008949, A007318.

Programs

Maple

T := (n, k) -> (-1)^k * binomial(n-1, k):
seq(print(seq(T(n, k), k = 0..n)), n = 0..7); # Peter Luschny, Apr 13 2023

Mathematica

T[0, 0] := 1;  T[n_, n_] := 0; T[n_, k_] := (-1)^k*Binomial[n - 1, k]; Table[T[n, k], {n, 0, 20}, {k, 0, n}] // Flatten (* G. C. Greubel, Aug 31 2017 *)

PARI

concat(1, for(n=1,10, for(k=0,n, print1(if(k != n, (-1)^k*binomial(n-1,k), 0), ", ")))) \\ G. C. Greubel, Aug 31 2017

Formula

T(n, 0) = 1, T(n, n) = 0^n, T(n, k) = -T(n-1, k-1) + T(n-1, k), for 0 < k < n.

T(n, k) = binomial(n-1, k)*(-1)^k, 0 <= k < n, T(n, n) = 0^n.

T(n, n-k-1) = -T(n, k), for 0 < k < n.

T(n, k) = A071919(n, k)*(-1)^k and A071919(n, k) = abs(T(n, k)).

Triangle T(n,k), 0 <= k <= n, read by rows, given by [1, 0, 0, 0, 0, 0, 0, 0, ...] DELTA [0, -1, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Sep 05 2005

G.f.: (1 + x*y) / (1 + x*y - x). - R. J. Mathar, Aug 11 2015

Extensions

Typo in name corrected by Andrey Zabolotskiy, Feb 22 2022

Offset corrected by Peter Luschny, Apr 13 2023