A059226 - OEIS

%I #22 Jan 07 2017 02:43:27

%S 1,0,1,0,2,4,0,4,12,18,0,8,32,70,94,0,16,80,224,426,544,0,32,192,648,

%T 1536,2708,3370,0,64,448,1760,4920,10596,17846,21878,0,128,1024,4576,

%U 14624,36552,74040,121014,146924,0,256,2304,11520,41248

%N Triangle T(n,k) (0 <= k <= n) read by rows: top entry is 1, all other rows begin with 0; typical entry is sum of entry to left plus sum of all entries above it in the triangle.

%C Variant of Boustrophedon transform applied to 1, 0, 0, 0, ...

%H Vincenzo Librandi, <a href="/A059226/b059226.txt">Rows n = 0..100, flattened</a>

%H <a href="/index/Bo#boustrophedon">Index entries for sequences related to boustrophedon transform</a>

%e Triangle begins:

%e   1;

%e   0,   1;

%e   0,   2,   4;

%e   0,   4,  12,  18;

%e   0,   8,  32,  70,  94;

%e   0,  16,  80, 224, 426, 544;

%e   ...

%e T(4,3) = 70 because it is the sum of the entry to the left (32) plus the sum of all the entries above position (4,3), which give 1 + 0 + 1 + 2 + 4 + 12 + 18.

%p T := proc(i,j) option remember; local r,s,t1; if i=0 and j=0 then RETURN(1); fi; if j=0 then RETURN(0); fi; t1 := T(i,j-1); for r from 0 to i-j do for s from 0 to j do if r+s <> i then t1 := t1+T(r+s,s); fi; od: od: RETURN(t1); end; # n-th row is T(n,0), T(n,1), ..., T(n,n)

%p To get the triangle formed when the left diagonal has a single 1 in position k:

%p T := proc(i,j,k) option remember; local r,s,t1; if i < k then RETURN(0); fi; if i = k then RETURN(1); fi; if j = 0 then RETURN(0); fi; t1 := T(i,j-1,k); for r from 0 to i-j do for s from 0 to j do if r+s <> i then t1 := t1+T(r+s,s,k); fi; od: od: t1; end;

%t T [i_, j_] := T[i, j] = Module[{r, s, t1}, If[i == 0 && j == 0, Return[1]]; If[j == 0, Return[0]]; t1 = T[i, j-1]; For[r = 0, r <= i-j, r++, For[s = 0, s <= j, s++, If[r+s != i, t1 = t1 + T[r+s, s]]]]; Return[t1]]; Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Dec 26 2013, translated from Maple *)

%Y Diagonals give A000079, A001787, A059224, A059229. Final entries in each row give A059227. Row sums give A059228. Cf. A059271.

%K nonn,easy,tabl,nice

%O 0,5

%A _N. J. A. Sloane_, Jan 19 2001