OFFSET
0,13
COMMENTS
Row sums are A078008. Diagonal sums are A078024. Inverse is A112466. Note that C(n,k) = sum{j = 0..n-k, C(j+k-1,j)}.
Central coefficients T(2n, n) are A072547. - Paul Barry, Apr 08 2011
T(n,k) = A108561(n, n-k). - Reinhard Zumkeller, Jan 03 2014
LINKS
Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
Roland Bacher, Chebyshev polynomials, quadratic surds and a variation of Pascal's triangle, arXiv:1509.09054 [math.CO], 2015.
E. Deutsch, L. Ferrari and S. Rinaldi, Production Matrices, Advances in Mathematics, 34 (2005) pp. 101-122.
FORMULA
Number triangle T(n, k) = sum{j = 0..n-k, C(j+k-1, j)*(-1)^(n-k-j)}.
T(n, 0) = (-1)^n, T(n, n) = 1, T(n+1, k) = T(n, k-1) + T(n, k), 0 < k < n. - Reinhard Zumkeller, Jan 03 2014
T(n, k) = T(n-1, k-1) + T(n-2,k) + T(n-2,k-1), T(0, 0) = 1, T(1, 0) = -1, T(1, 1) = 1, T(n, k) = 0 if k < 0 or if k > n. - Philippe Deléham, Jan 11 2014
exp(x) * e.g.f. for row n = e.g.f. for diagonal n. For example, for n = 3 we have exp(x)*(-1 + x + x^2/2! + x^3/3!) = -1 + 2*x^2/2! + 6*x^3/3! + 13*x^4/4! + .... The same property holds more generally for Riordan arrays of the form ( f(x), x/(1 - x) ). - Peter Bala, Dec 21 2014
EXAMPLE
Triangle starts
1;
-1,1;
1,0,1;
-1,1,1,1;
1,0,2,2,1;
-1,1,2,4,3,1;
1,0,3,6,7,4,1;
Production matrix begins
-1, 1,
0, 1, 1,
0, 0, 1, 1,
0, 0, 0, 1, 1,
0, 0, 0, 0, 1, 1,
0, 0, 0, 0, 0, 1, 1,
0, 0, 0, 0, 0, 0, 1, 1,
0, 0, 0, 0, 0, 0, 0, 1, 1
- Paul Barry, Apr 08 2011
MATHEMATICA
T[n_, k_] := Sum[Binomial[j + k - 1, j]*(-1)^(n - k - j), {j, 0, n - k}];
Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 23 2018 *)
PROG
(Haskell)
a112465 n k = a112465_tabl !! n !! k
a112465_row n = a112465_tabl !! n
a112465_tabl = iterate f [1] where
f xs'@(x:xs) = zipWith (+) ([-x] ++ xs ++ [0]) ([0] ++ xs')
-- Reinhard Zumkeller, Jan 03 2014
CROSSREFS
KEYWORD
AUTHOR
Paul Barry, Sep 06 2005
STATUS
approved