A145975 Triangle read by rows, partition triangle A027293 convolved with A010815.

1, 1, -1, 2, -1, -1, 3, -2, -1, 0, 5, -3, -2, 0, 0, 7, -5, -3, 0, 0, 1, 11, -7, -5, 0, 0, 1, 0, 15, -11, -7, 0, 0, 2, 0, 1, 22, -15, -11, 0, 0, 3, 0, 1, 0, 30, -22, -15, 0, 0, 5, 0, 2, 0, 0, 42, -30, -22, 0, 0, 7, 0, 3, 0, 0, 0, 56, -42, -30, 0, 0, 11, 0, 5, 0, 0, 0, 0

Offset

1,4

Comments

Row sums = [1, 0, 0, 0,...]. (a set of matrix operations equivalent to the comment in A010815 that "convolved with the partition numbers = [1, 0, 0, 0,...].

Links

Robert Price, Table of n, a(n) for n = 1..1275 (first 50 rows)

Formula

Triangle read by rows, = (A027293 * (A010815 * 0^(n-k)); 0<=k<=n. A027293 = an infinite lower triangular matrix with A000041 in every column (the partition numbers). A010815 = (1, -1, -1, 0, 0, 1,...)

Example

First few rows of the triangle =
1;
1, -1;
2, -1, -1;
3, -2, -1, 0;
5, -3, -2, 0, 0;
7, -5, -3, 0, 0, 1;
11, -7, -5, 0, 0, 1, 0;
15, -11, -7, 0, 0, 2, 0, 1;
22, -15, -11, 0, 0, 3, 0, 1, 0;
30, -22, -15, 0, 0, 5, 0, 2, 0, 0;
42, -30, -22, 0, 0, 7, 0, 3, 0, 0, 0;
...

Mathematica

Table[Count[Flatten[Union /@ IntegerPartitions@n], k]*SeriesCoefficient[Product[1 - x^i, {i, k - 1}], {x, 0, k - 1}], {n, 12}, {k, n}] // Flatten (* Robert Price, Jun 15 2020 *)

Crossrefs

A027293, Cf. A010815, A000041

Sequence in context: A127948 A177350 A190688 * A211028 A239001 A277648

Adjacent sequences: A145972 A145973 A145974 * A145976 A145977 A145978

Keyword

tabl,sign

Author

Gary W. Adamson, Oct 25 2008

Extensions

Missing zero at a(55) inserted by Robert Price, Jun 15 2020

Status

approved