OFFSET
0,12
COMMENTS
Apart from initial term, same as A112467. - Philippe Deléham, Nov 07 2006
From Christopher Simoni, Feb 11 2019: (Start)
Along the diagonal containing all the natural numbers, where every number is at point T(n,n-1), T(n,k)^2 can be found by adding T(n+1,k), T(n,k-1), T(n-1,k), and T(n,k+1) together. Empirical observation.
Along the diagonal containing all the natural numbers, where every number is at point T(n,n-1), T(n,k)^3 can by found. The following products are added together: T(n-1,k-1)*T(n,k+1)*T(n+1,k); T(n-1,k)*T(n+1,k+1)*(n,k-1); and 3*T(n,k). Empirical observation. (End)
REFERENCES
A. A. Kirillov, Variations on the triangular theme, Amer. Math. Soc. Transl., (2), Vol. 169, 1995, pp. 43-73, see p. 71.
LINKS
G. C. Greubel, Rows n=0..100 of triangle, flattened
I. Gessel and S. Ree, Lattice paths and Faber polynomials, Advances in Combinatorial Methods and Applications to Probability and Statistics pp 3-13. DOI: 10.1007/978-1-4612-4140-9_1.
FORMULA
T(n, k) = T(n-1, k-1) + T(n-1, k); T(0, 0) = 0, T(1, 0) = -1, T(1, 1) = 1. - Michael Somos, Oct 04 1999
T(n, k) = binomial(n, k-1) - binomial(n, k). - Michael Somos, Oct 04 1999
T(2n+1,n+1) = A000108(n), Catalan numbers. - Philippe Deléham, Nov 29 2013
a(n) = -A037012(n). - Michael Somos, May 24 2015
T(n, k) = -T(n, n - k); T(n, n) = 1 except for n = 0; T(n+1, n) = n-1; T(n+2, n) = A000217(n-1) - 1. This proves the observations by C. Simoni, e.g., T(n+1,n)^2 - 1 = T(n+1,n-1) + T(n+2,n) + 1. - M. F. Hasler, Feb 11 2019
EXAMPLE
Triangle begins:
0
-1 1
-1 0 1
-1 -1 1 1
-1 -2 0 2 1
-1 -3 -2 2 3 1
-1 -4 -5 0 5 4 1
-1 -5 -9 -5 5 9 5 1
-1 -6 -14 -14 0 14 14 6 1
-1 -7 -20 -28 -14 14 28 20 7 1
-1 -8 -27 -48 -42 0 42 48 27 8 1
-1 -9 -35 -75 -90 -42 42 90 75 35 9 1
-1 -10 -44 -110 -165 -132 0 132 165 110 44 10 1
... - Philippe Deléham, Nov 29 2013
MATHEMATICA
Table[Binomial[n-1, k-1] -Binomial[n-1, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Feb 11 2019 *)
PROG
(PARI) {T(n, k) = if( n<1, 0, polcoeff( (x-1) * (1+x)^(n-1), k))}; /* Michael Somos, Oct 04 1999 */
(PARI) A008482(n, k)={if(k>=n-k, if(k>n-k, -A008482(n, n-k), 0), k>2, A008482(n-1, k-1)+A008482(n-1, k), k>1, 1-(n-2)*(n-3)\2, k, 2-n, -1)} \\ M. F. Hasler, Feb 11 2019
(Sage) [0] + flatten([[binomial(n-1, k-1) -binomial(n-1, k) for k in (0..n)] for n in (1..12)]) # G. C. Greubel, Feb 11 2019
CROSSREFS
KEYWORD
AUTHOR
STATUS
approved