%I A052509
%S A052509 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,
%T A052509 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,
%U A052509 16,8,4,2,1,1,11,46,93,99,63,32,16,8,4,2,1,1,12,56,130,163,120,64,32
%N 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.
%C A052509 Also square array T(n,k) (n >= 0, k >= 0) read by antidiagonals: T(n,
               k) = Sum_{i=0..k} C(n,i).
%C A052509 As a number triangle read by rows, this is T(n,k)=sum{i=n-2k..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 (pbarry(AT)wit.ie), Sep 04 2005
%H A052509 <a href="Sindx_Pas.html#Pascal">Index entries for triangles and arrays 
               related to Pascal's triangle</a>
%F A052509 T(n, m)= Sum( k=0..n, C(n-m, m-k) ) - Wouter Meeussen (wouter.meeussen(AT)pandora.be), 
               Oct 03 2002
%e A052509 Rows: {1}, {1,1}, {1,2,1}, {1,3,2,1}, {1,4,4,2,1}, ...
%e A052509 Triangle begins:
%e A052509 1
%e A052509 1,1
%e A052509 1,2,1
%e A052509 1,3,2,1
%e A052509 1,4,4,2,1
%e A052509 1,5,7,4,2,1
%e A052509 1,6,11,8,4,2,1
%e A052509 As a square array, this begins:
%e A052509 1 1 1 1 1 1 ...
%e A052509 1 2 2 2 2 2 ...
%e A052509 1 3 4 4 4 4 ...
%e A052509 1 4 7 8 8 8 ...
%e A052509 1 5 11 15 16 ...
%e A052509 1 6 16 26 31 32 ...
%p A052509 a := proc(n::nonnegint, k::nonnegint) option remember: if k=0 then RETURN(1) 
               fi:
%p A052509 if k=n then RETURN(1) fi: a(n-1,k)+a(n-2,k-1) end:for n from 0 to 20 
               do
%p A052509 for k from 0 to n do printf(`%d,`,a(n,k)) od:od:
%p A052509 with(combinat): for s from 0 to 20 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
%t A052509 Table[Sum[Binomial[n-m, m-k], {k, 0, n}], {n, 0, 10}, {m, 0, n}]
%Y A052509 Cf. A054123, A054124, A007318, A008949.
%Y A052509 Row sums = Fibonacci numbers - 1.
%Y A052509 Columns give A000027, A000124, A000125, A000129, A006261, ...
%Y A052509 Cf. A052509, A054123, A054124, A007318, A008949, A052553.
%Y A052509 Partial sums across rows of (extended) Pascal's triangle A052553.
%Y A052509 Sequence in context: A077592 A055794 A092905 this_sequence A093628 A114282 
               A112739
%Y A052509 Adjacent sequences: A052506 A052507 A052508 this_sequence A052510 A052511 
               A052512
%K A052509 nonn,tabl,easy,nice
%O A052509 0,5
%A A052509 N. J. A. Sloane (njas(AT)research.att.com), Mar 17, 2000
%E A052509 More terms and Maple code from James A. Sellers (sellersj(AT)math.psu.edu), 
               Mar 17 2000
%E A052509 Entry formed by merging two earlier entries. Formulae probably need editing. 
               - N. J. A. Sloane (njas(AT)research.att.com), Jun 17 2007