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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

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:

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 and Maple code 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

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Last modified December 21 16:36 EST 2014. Contains 252324 sequences.