A026568 Irregular triangular array T read by rows: T(i,0) = T(i,2i) = 1 for i >= 0; T(i,1) = T(i,2i-1) = [ (i+1)/2 ] for i >= 1; and for i >= 2 and 2 <=j <= i - 2, T(i,j) = T(i-1,j-2) + T(i-1,j-1) + T(i-1,j) if i + j is even, T(i,j) = T(i-1,j-2) + T(i-1,j) if i + j is odd.

1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 2, 4, 5, 4, 2, 1, 1, 2, 7, 7, 13, 7, 7, 2, 1, 1, 3, 8, 16, 20, 27, 20, 16, 8, 3, 1, 1, 3, 12, 19, 44, 43, 67, 43, 44, 19, 12, 3, 1, 1, 4, 13, 34, 56, 106, 111, 153, 111, 106, 56, 34, 13, 4, 1, 1, 4, 18, 38, 103, 140, 273

Offset: 1

Author

Clark Kimberling

Keywords

• nonn
• tabf

Comments

T(n, k) = number of strings s(0)..s(n) such that s(0) = 0, s(n) = n - k, |s(i)-s(i-1)| <= 1 if s(i-1) is even, |s(i)-s(i-1)| = 1 if s(i-1) is odd, for 1 <= i <= n.

Examples

			First 5 rows:
  1
  1  1  1
  1  1  3  1  1
  1  2  4  5  4  2  1
  1  2  7  7 13  7  7  2  1
		

Links

• Clark Kimberling, Rows 0..100, flattened
• Index entries for triangles and arrays related to Pascal's triangle

Crossrefs

Cf. A026519, A026536, A026552, A026584, A027926.

Cf. T(n,n) is A026569.

Programs

Mathematica

z = 12; t[n_, 0] := 1; t[n_, 1] := Floor[(n + 1)/2]; t[n_, k_] := t[n, k] = Which[k == 2 n, 1, k == 2 n - 1, Floor[(n + 1)/2], EvenQ[n + k], t[n - 1, k - 2] + t[n - 1, k - 1] + t[n - 1, k], OddQ[n + k], t[n - 1, k - 2] + t[n - 1, k]]; u = Table[t[n, k], {n, 0, z}, {k, 0, 2 n}];
TableForm[u] (* A026568 array *)
Flatten[u]   (* A026568 sequence *)

PARI

T(k,n)=if(n<0||n>2*k,0,if(n==0||n==2*k,1,if(k>0&&(n==1||n==2*k-1),(k+1)\2,T(k-1,n-2)+T(k-1,n)+if((k+n)%2==0,T(k-1,n-1))))) \\ Ralf Stephan

Extensions

Updated by Clark Kimberling, Aug 28 2014