A175010 Triangle generated from INVERT transforms of variants of A080995.

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 1, 1, 2, 5, 1, 1, 1, 1, 1, 4, 6, 1, 1, 1, 1, 1, 2, 6, 9, 1, 1, 1, 1, 1, 1, 4, 8, 12, 1, 1, 1, 1, 1, 1, 2, 6, 12, 16, 1, 1, 1, 1, 1, 1, 1, 4, 8, 19, 18, 1, 1, 1, 1, 1, 1, 1, 2, 6, 11, 28, 23

Offset: 1

Author

Gary W. Adamson, Apr 03 2010

Keywords

• nonn
• tabl

Comments

Row sums = A000041 starting with offset 1: (1, 1, 2, 3, 5, 7, 11, 15, ...).

The INVERTi transform of A000041 starting with offset 1 follows from the definition of the INVERT transform, given 1/p(x) = A010815.

Examples

			First few rows of the array:
1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101,
1, 1, 2, 3, 4,  6,  9, 13, 18, 26, 38, 54,  76,
1, 1, 1, 2, 3,  4,  5,  7, 10, 14, 19, 26,  35,
1, 1, 1, 1, 2,  3,  4,  5,  6,  8, 11, 15,  20,
1, 1, 1, 1, 1,  2,  3,  4,  5,  6,  7,  9,  12,
1, 1, 1, 1, 1,  1,  2,  3,  4,  5,  6,  7,   8,
...
Taking finite differences from the bottom starting with the top "1", we obtain rows of the triangle:
1;
1, 1;
1, 1, 1;
1, 1, 1, 2;
1, 1, 1, 1, 3;
1, 1, 1, 1, 2, 5;
1, 1, 1, 1, 1, 4, 6;
1, 1, 1, 1, 1, 2, 6, 9;
1, 1, 1, 1, 1, 1, 4, 8, 12;
1, 1, 1, 1, 1, 1, 2, 6, 12, 16;
1, 1, 1, 1, 1, 1, 1, 4,  8, 19, 18;
1, 1, 1, 1, 1, 1, 1, 2,  6, 11, 28, 23;
1, 1, 1, 1, 1, 1, 1, 1,  4,  8, 15, 41, 25;
1, 1, 1, 1, 1, 1, 1, 1,  2,  6, 10, 22, 61, 26;
...
Example: Row 2 = INVERT transform of Q(x^2), (i.e., Q(x) interleaved with one zero between terms).
		

Crossrefs

Cf. A000041, A080995, A010815.

Formula

Given the INVERTi transform of the partition numbers starting with offset 1 = a signed variant of A080995 such that Q = (1, 1, 0, 0, -1, 0, -1, 0, 0, 0, 0, 1, ...).

Construct an array in which k-th row (k=1,2,3,...) = the INVERT transform of Q(x^k), i.e., where polcoeff Q(x) is interleaved with 0,1,2,3,... zeros.

Take finite differences of the array terms starting with the last "1" going from the bottom to top, becoming rows of triangle A175010.