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A052509
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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.
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14
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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; internal format)
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OFFSET
| 0,5
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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
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LINKS
| Index entries for triangles and arrays related to Pascal's triangle
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FORMULA
| T(n, k) = sum(m=0..n, C(n-k, k-m) ) - Wouter Meeussen, Oct 03 2002
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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 ...
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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 (sellersj(AT)math.psu.edu), Mar 17 2000
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MATHEMATICA
| Table[Sum[Binomial[n-k, k-m], {m, 0, n}], {n, 0, 10}, {k, 0, n}]
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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 */
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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 A093628 A186807
Adjacent sequences: A052506 A052507 A052508 * A052510 A052511 A052512
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KEYWORD
| nonn,tabl,easy,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Mar 17, 2000
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EXTENSIONS
| More terms and Maple code from James A. Sellers (sellersj(AT)math.psu.edu), Mar 17 2000
Entry formed by merging two earlier entries. - N. J. A. Sloane (njas(AT)research.att.com), Jun 17 2007
Edited by Johannes W. Meijer, Jul 24 2011
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