A145573 - OEIS

A145573

Characteristic partition array for partitions without part 1.

0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 0, 0, 0

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OFFSET

1,1

COMMENTS

The partitions are ordered according to Abramowitz-Stegun (A-St order). See e.g. A036040 for the reference, pp. 831-2.

The row lengths of this array are p(n)=A000041(n) (number of partitions of n).

The entries of row n are grouped together for partitions with rising parts number m from 1 to n. The number of partitions of n with m parts is p(n,m)= A008284(n,m), m=1..n, n>=1.

For the array without zeros see A145574.

LINKS

Table of n, a(n) for n=1..105.

W. Lang and M. Sjodahl First 10 rows of the array and row sums.

FORMULA

As array: a(n,k)=1 if the k-th partition of n in A-St order has no part 1, and a(n,k)=0 else.

Translated into the sequence a(m) entry: a(n,k) = a(sum(p(k),k=1..n)+k).

EXAMPLE

[0],[1,0],[1,0,0],[1,0,1,0,0],[1,0,1,0,0,0,0],...

a(4,3) = a(1+2+3+3) = a(9) = 1 because a(4,3) belongs to the partition [2^2]=[2,2] of n=4 which has no part 1.

CROSSREFS

Cf. A145574 (without zeros). A002865 (row sums).