login
This site is supported by donations to The OEIS Foundation.

 

Logo

Please make a donation to keep the OEIS running. We are now in our 55th year. In the past year we added 12000 new sequences and reached 8000 citations (which often say "discovered thanks to the OEIS"). We need to raise money to hire someone to manage submissions, which would reduce the load on our editors and speed up editing.
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A112465 Riordan array (1/(1+x),x/(1-x)). 5
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 (list; table; graph; refs; listen; history; text; internal format)
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 A086275

Adjacent sequences:  A112462 A112463 A112464 * A112466 A112467 A112468

KEYWORD

easy,sign,tabl

AUTHOR

Paul Barry, Sep 06 2005

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 10 18:10 EST 2019. Contains 329901 sequences. (Running on oeis4.)