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

 

Logo

Annual Appeal: Please make a donation (tax deductible in USA) to keep the OEIS running. Over 5000 articles have referenced us, often saying "we discovered this result with the help of the OEIS".

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A052509 Knights-move Pascal triangle: T(n,k), n >= 0, 0 <= k <= n; T(n,0) = T(n,n) = 1, T(n,k) = T(n-1,k) + T(n-2,k-1) for k = 1,2,...,n-1, n >= 2. 17
1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 4, 4, 2, 1, 1, 5, 7, 4, 2, 1, 1, 6, 11, 8, 4, 2, 1, 1, 7, 16, 15, 8, 4, 2, 1, 1, 8, 22, 26, 16, 8, 4, 2, 1, 1, 9, 29, 42, 31, 16, 8, 4, 2, 1, 1, 10, 37, 64, 57, 32, 16, 8, 4, 2, 1, 1, 11, 46, 93, 99, 63, 32, 16, 8, 4, 2, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

Also square array T(n,k) (n >= 0, k >= 0) read by antidiagonals: T(n,k) = Sum_{i=0..k} C(n,i).

As a number triangle read by rows, this is T(n,k) = sum{i=n-2*k..n-k, binomial(n-k,i)}, with T(n,k) = T(n-1,k) + T(n-2,k-1). Row sums are A000071(n+2). Diagonal sums are A023435(n+1). It is the reverse of the Whitney triangle A004070. - Paul Barry, Sep 04 2005

LINKS

Reinhard Zumkeller, Rows n = 0..150 of triangle, flattened

D. J. Price, Some unusual series occurring in n-dimensional geometry, Math. Gaz., 30 (1946), 149-150.

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

FORMULA

T(n, k) = sum(m=0..n, C(n-k, k-m) ). - Wouter Meeussen, Oct 03 2002

EXAMPLE

Triangle begins:

1

1,1

1,2,1

1,3,2,1

1,4,4,2,1

1,5,7,4,2,1

1,6,11,8,4,2,1

As a square array, this begins:

1 1 1 1 1 1 ...

1 2 2 2 2 2 ...

1 3 4 4 4 4 ...

1 4 7 8 8 8 ...

1 5 11 15 16 ...

1 6 16 26 31 32 ...

MAPLE

a := proc(n::nonnegint, k::nonnegint) option remember: if k=0 then RETURN(1) fi: if k=n then RETURN(1) fi: a(n-1, k)+a(n-2, k-1) end: for n from 0 to 11 do for k from 0 to n do printf(`%d, `, a(n, k)) od: od: # James A. Sellers, Mar 17 2000

with(combinat): for s from 0 to 11 do for n from s to 0 by -1 do if n=0 or s-n=0 then printf(`%d, `, 1) else printf(`%d, `, sum(binomial(n, i), i=0..s-n)) fi; od: od: # James A. Sellers, Mar 17 2000

MATHEMATICA

Table[Sum[Binomial[n-k, k-m], {m, 0, n}], {n, 0, 10}, {k, 0, n}]

PROG

(PARI) T(n, k)=sum(m=0, n, binomial(n-k, k-m));

for(n=0, 10, for(k=0, n, print1(T(n, k), ", "); ); print(); ); /* show triangle */

(Haskell)

a052509 n k = a052509_tabl !! n !! k

a052509_row n = a052509_tabl !! n

a052509_tabl = [1] : [1, 1] : f [1] [1, 1] where

   f row' row = rs : f row rs where

     rs = zipWith (+) ([0] ++ row' ++ [1]) (row ++ [0])

-- Reinhard Zumkeller, Nov 22 2012

CROSSREFS

Cf. A054123, A054124, A007318, A008949.

Row sums A000071; Diagonal sums A023435; Mirror A004070.

Columns give A000027, A000124, A000125, A000127, A006261, ...

Cf. A052509, A054123, A054124, A007318, A008949, A052553.

Partial sums across rows of (extended) Pascal's triangle A052553.

Sequence in context: A194005 A055794 A092905 * A172119 A228125 A227588

Adjacent sequences:  A052506 A052507 A052508 * A052510 A052511 A052512

KEYWORD

nonn,tabl,easy,nice

AUTHOR

N. J. A. Sloane, Mar 17 2000

EXTENSIONS

More terms from James A. Sellers, Mar 17 2000

Entry formed by merging two earlier entries. - N. J. A. Sloane, Jun 17 2007

Edited by Johannes W. Meijer, Jul 24 2011

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 December 9 06:44 EST 2016. Contains 278963 sequences.