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

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A122542 Triangle T(n,k), 0<=k<=n, read by rows, given by [0, 2, -1, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. 19
1, 0, 1, 0, 2, 1, 0, 2, 4, 1, 0, 2, 8, 6, 1, 0, 2, 12, 18, 8, 1, 0, 2, 16, 38, 32, 10, 1, 0, 2, 20, 66, 88, 50, 12, 1, 0, 2, 24, 102, 192, 170, 72, 14, 1, 0, 2, 28, 146, 360, 450, 292, 98, 16, 1, 0, 2, 32, 198, 608, 1002, 912, 462, 128, 18, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

Riordan array (1, x*(1+x)/(1-x)) . Rising and falling diagonals are the tribonacci numbers A000213, A001590.

LINKS

Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened

Bela Bajnok, Additive Combinatorics: A Menu of Research Problems, arXiv:1705.07444 [math.NT], May 2017. See Sect. 2.3.

FORMULA

Sum_{k=0..n} x^k*T(n,k) = A000007(n), A001333(n), A104934(n), A122558(n), A122690(n), A091928(n)  for x = 0, 1, 2, 3, 4, 5 . Philippe Deléham, Jan 25 2012 .

Sum_{k=0..n} 3^(n-k)*T(n,k) = A086901(n).

Sum_{k=0..n} 2^(n-k)*T(n,k) = A007483(n-1), n>=1 . - Philippe Deléham, Oct 08 2006

T(2*n,n) = A123164(n).

T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k-1), n>1. - Philippe Deléham, Jan 25 2012

G.f.: (1-x)/(1-(1+y)*x-y*x^2). - Philippe Deléham, Mar 02 2012

EXAMPLE

Triangle begins:

1;

0, 1;

0, 2, 1;

0, 2, 4, 1;

0, 2, 8, 6, 1;

0, 2, 12, 18, 8, 1;

0, 2, 16, 38, 32, 10, 1;

0, 2, 20, 66, 88, 50, 12, 1;

0, 2, 24, 102, 192, 170, 72, 14, 1;

0, 2, 28, 146, 360, 450, 292, 98, 16, 1;

0, 2, 32, 198, 608, 1002, 912, 462, 128, 18, 1;

PROG

(Haskell)

a122542 n k = a122542_tabl !! n !! k

a122542_row n = a122542_tabl !! n

a122542_tabl = map fst $ iterate

   (\(us, vs) -> (vs, zipWith (+) ([0] ++ us ++ [0]) $

                      zipWith (+) ([0] ++ vs) (vs ++ [0]))) ([1], [0, 1])

-- Reinhard Zumkeller, Jul 20 2013, Apr 17 2013

(Sage)

def A122542_row(n):

    @cached_function

    def prec(n, k):

        if k==n: return 1

        if k==0: return 0

        return prec(n-1, k-1)+2*sum(prec(n-i, k-1) for i in (2..n-k+1))

    return [prec(n, k) for k in (0..n)]

for n in (0..10): print A122542_row(n) # Peter Luschny, Mar 16 2016

CROSSREFS

Other versions: A035607, A113413, A119800, A266213.

Diagonals : A000012, A005843, A001105, A035597-A035606.

Columns : A000007, A040000, A008575, A005899, A008412-A008416, A008418, A008420, A035706-A035745.

Cf. A155161, A059283.

Sequence in context: A206022 A115247 A204163 * A227341 A098542 A141343

Adjacent sequences:  A122539 A122540 A122541 * A122543 A122544 A122545

KEYWORD

nonn,tabl

AUTHOR

Philippe Deléham, Sep 19 2006, May 28 2007

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 | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified September 20 16:14 EDT 2017. Contains 292276 sequences.