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

0,8

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

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

Example

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.

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 *)

Prog

(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

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.

Sequence in context: A119337 A213889 A363779 * A097805 A071919 A321791

Adjacent sequences: A110552 A110553 A110554 * A110556 A110557 A110558

Keyword

sign,easy,tabl

Author

Reinhard Zumkeller, Jul 27 2005

Extensions

Typo in name corrected by Andrey Zabolotskiy, Feb 22 2022

Offset corrected by Peter Luschny, Apr 13 2023

Status

approved