A059226 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.

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, 1536, 2708, 3370, 0, 64, 448, 1760, 4920, 10596, 17846, 21878, 0, 128, 1024, 4576, 14624, 36552, 74040, 121014, 146924, 0, 256, 2304, 11520, 41248

Offset

0,5

Comments

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

Links

Vincenzo Librandi, Rows n = 0..100, flattened

Index entries for sequences related to boustrophedon transform

Example

Triangle begins:
  1;
  0,   1;
  0,   2,   4;
  0,   4,  12,  18;
  0,   8,  32,  70,  94;
  0,  16,  80, 224, 426, 544;
  ...
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.

Maple

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

Mathematica

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 *)

Crossrefs

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

Sequence in context: A341862 A256487 A079985 * A221655 A221087 A279580

Adjacent sequences: A059223 A059224 A059225 * A059227 A059228 A059229

Keyword

nonn,easy,tabl,nice

Author

N. J. A. Sloane, Jan 19 2001

Status

approved