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%I #16 Mar 02 2015 16:08:45
%S 1,1,2,5,15,55,239,1199,6810,43108,300731,2291162,18923688,168402163,
%T 1606199354,16345042652,176758631046,2024225038882,24471719797265,
%U 311446235344127,4162172487402027,58275220793611957,853045299274146032
%N Variation of Boustrophedon transform applied to sequence 1,0,0,0,...: fill an array by diagonals in alternating directions - 'up' and 'down'. The first element of each diagonal after the first is 0. When 'going up', add to the previous element the elements of the row the new element is in. When 'going down', add to the previous element the elements of the column the new element is in. The final element of the n-th diagonal is a(n).
%H Vincenzo Librandi, <a href="/A059219/b059219.txt">Table of n, a(n) for n = 0..120</a>
%H <a href="/index/Bo#boustrophedon">Index entries for sequences related to boustrophedon transform</a>
%e The array begins
%e 1 1 0 5 0 55 0 ...
%e 0 1 3 5 48 55 ...
%e 2 2 8 39 103 ...
%e 0 12 27 152 ...
%e 15 15 190 ...
%e 0 221 ...
%p aaa := proc(m,n) option remember; local j,s,t1; if m=0 and n=0 then RETURN(1); fi; if n = 0 and m mod 2 = 1 then RETURN(0); fi; if m = 0 and n mod 2 = 0 then RETURN(0); fi; s := m+n; if s mod 2 = 1 then t1 := aaa(m+1,n-1); for j from 0 to n-1 do t1 := t1+aaa(m,j); od: else t1 := aaa(m-1,n+1); for j from 0 to m-1 do t1 := t1+aaa(j,n); od: fi; RETURN(t1); end; # the n-th antidiagonal in the up direction is aaa(n,0), aaa(n-1,1), aaa(n-2,2), ..., aaa(0,n)
%t max = 22; t[0, 0] = 1; t[0, _?EvenQ] = 0; t[_?OddQ, 0] = 0; t[n_, k_] /; OddQ[n+k](* up *):= t[n, k] = t[n+1, k-1] + Sum[t[n, j], {j, 0, k-1}]; t[n_, k_] /; EvenQ[n+k](* down *):= t[n, k] = t[n-1, k+1] + Sum[t[j, k], {j, 0, n-1}]; tnk = Table[t[n, k], {n, 0, max}, {k, 0, max-n}]; Join[{1}, Rest[Union[tnk[[1]], tnk[[All, 1]]]]](* _Jean-François Alcover_, May 16 2012 *)
%Y Cf. A000667, A059216, A059217, A059220, A059237, A059720, A059718.
%K easy,nonn,nice
%O 0,3
%A _N. J. A. Sloane_, Jan 18 2001
%E More terms from _Floor van Lamoen_, Jan 19 2001; and from _N. J. A. Sloane_ Jan 20 2001.
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