A173239 Triangle by columns, A000041 shifted down thrice, k>=0.

1, 1, 2, 3, 1, 5, 1, 7, 2, 11, 3, 1, 15, 5, 1, 22, 7, 2, 30, 11, 3, 1, 42, 15, 5, 1, 56, 22, 7, 2, 77, 30, 11, 3, 1, 101, 42, 15, 5, 1, 135, 56, 22, 7, 2, 176, 77, 30, 11, 3, 1, 231, 101, 42, 15, 5, 1, 297, 135, 56, 22, 7, 2, 385, 176, 77, 30, 11, 3, 1

Offset: 0

Author

Gary W. Adamson, Feb 13 2010

Keywords

• nonn
• tabf
• easy

Comments

Row sums = A024787, the numbers of 3's in all partitions of n, where A024787 starts with offset 1: (0, 0, 1, 1, 2, 4, 6, 9, 15,...). Triangle A173239 row sums start with the first "1" of A024787.

Let the triangle = M as an infinite lower triangular matrix. Then Lim_{n->inf} = A173241, the Euler transform of A051064 (the ruler function for 3).

Let P(x) = polcoeff A000041 = (1 + x + 2x^2 + 3x^3 + 5x^4 + 7x^5 + ...), then P(x) = A(x) / A(x^3), where A(x) = polcoeff A173241: (1 + x + 2x^2 + 4x^3 + 6x^4 + ...)

Refer to A173238 comments for three conjectures relating A000041 to the infinite set of generalized ruler function sequences.

Examples

			First few rows of the triangle =
1;
1;
2;
3, 1;
5, 1;
7, 2;
11, 3, 1;
15, 5, 1;
22, 7, 2;
30, 11, 3, 1;
42, 15, 5, 1;
56, 22, 7, 2;
77, 30, 11, 3, 1;
101, 42, 15, 5, 1;
135, 56, 22, 7, 2;
176, 77, 30, 11, 3, 1;
231, 101, 42, 15, 5, 1;
297, 135, 56, 22, 7, 2;
385, 176, 77, 30, 11, 3, 1;
490, 231, 101, 42, 15, 5, 1;
627, 297, 135, 56, 22, 7, 2;
792, 385, 176, 77, 30, 11, 3, 1;
1002,490, 231, 101, 42, 15, 5, 1;
1255, 627, 297, 135, 56, 22, 7, 2;
1575, 792, 385, 176, 77, 30, 11, 3, 1;
...
		

Crossrefs

Cf. A000041, A173238, A173241, A051064, A024787.

Formula

T(n,k) = A000041(n-3*k) for k=0..floor(n/3).