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

Author

N. J. A. Sloane, Jan 19 2001

Keywords

• nonn
• easy
• tabl
• nice

Comments

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

Examples

			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.
		

Links

• Vincenzo Librandi, Rows n = 0..100, flattened
• Index entries for sequences related to boustrophedon transform

Crossrefs

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

Programs

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 *)