A112465 Riordan array (1/(1+x),x/(1-x)).

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, -1, 1, 3, 9, 13, 11, 5, 1, 1, 0, 4, 12, 22, 24, 16, 6, 1, -1, 1, 4, 16, 34, 46, 40, 22, 7, 1, 1, 0, 5, 20, 50, 80, 86, 62, 29, 8, 1, -1, 1, 5, 25, 70, 130, 166, 148, 91, 37, 9, 1, 1, 0, 6, 30, 95, 200, 296, 314, 239, 128, 46, 10, 1

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.

Index entries for triangles and arrays related to Pascal's triangle

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

Cf. A112468, A059260.

Sequence in context: A031282 A085685 A267632 * A112468 A207194 A349670

Adjacent sequences: A112462 A112463 A112464 * A112466 A112467 A112468

Keyword

easy,sign,tabl

Author

Paul Barry, Sep 06 2005

Status

approved