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

1,10

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.

Links

Table of n, a(n) for n=1..78.

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.

Example

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.

Sequence in context: A158464 A056926 A137773 * A107454 A371212 A250261

Adjacent sequences: A175007 A175008 A175009 * A175011 A175012 A175013

Keyword

nonn,tabl

Author

Gary W. Adamson, Apr 03 2010

Status

approved