A245618 Triangle {H(n,k)} similar to Pascal's with sides of 1's, but interior entries are obtained by the rule: H(n,k) = |H(n-1,k)+(-1)^m(n,k)*H(n-1,k-1)|, where m(n,k) = H(n-1,k) + H(n-1,k-1).

1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 4, 4, 1, 1, 1, 2, 3, 8, 3, 2, 1, 1, 1, 1, 5, 5, 1, 1, 1, 1, 2, 2, 6, 10, 6, 2, 2, 1, 1, 1, 4, 8, 16, 16, 8, 4, 1, 1, 1, 2, 3, 12, 24, 32, 24, 12, 3, 2, 1, 1, 1, 1, 9, 36, 56, 56, 36, 9, 1, 1, 1, 1, 2, 2, 10

Offset: 0

Author

Vladimir Shevelev, Nov 05 2014

Keywords

• nonn
• tabl

Comments

Let us consider the operation <+> over integers such that k<+>m = |k+(-1)^(k+m)*m|. Then H(n,k) = H(n-1,k)<+>H(n-1,k-1).

This is an analog of the formula binomial(n,k) = binomial(n-1,k) + binomial(n-1,k-1).

Examples

			Triangle begins
1
1  1
1  2  1
1  1  1  1
1  2  2  2  1
1  1  4  4  1  1
1  2  3  8  3  2  1
....................
		

Links

• Peter J. C. Moses, First 50 rows.

Crossrefs

Cf. A007318, row sums in A245619, row "sums", using <+>, in A249388.

Programs

Mathematica

parityAdd[a_, b_] := Abs[a + b (-1)^(a + b)];
triangleHP[n_, 0] := 1;
triangleHP[n_, n_] := 1;
triangleHP[n_, k_] := triangleHP[n, k] = parityAdd[triangleHP[n - 1, k - 1], triangleHP[n - 1, k]];
Flatten[Table[triangleHP[n, k], {n, 0, 15}, {k, 0, n}]] (* Peter J. C. Moses, Nov 05 2014 *)

Extensions

More terms from Peter J. C. Moses, Nov 05 2014