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.
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.
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.
KEYWORD
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