A145362 - OEIS

A145362

Lower triangular array, called S1hat(-1), related to partition number array A145361.

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

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OFFSET

1,1

COMMENTS

If in the partition array M31hat(-1):=A145361 entries belonging to partitions with the same parts number m are summed one obtains this triangle of numbers S1hat(-1). In the same way the signless Stirling1 triangle |A008275| is obtained from the partition array M_2 = A036039.

The first column is [1,1,0,0,0,...]=A008279(1,n-1), n>=1.

a(n,m) gives the number of partitions of n with m parts, with each part not exceeding 2. - Wolfdieter Lang, Aug 03 2023

LINKS

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

Wolfdieter Lang, First 10 rows of the array and more.

Wolfdieter Lang, Combinatorial Interpretation of Generalized Stirling Numbers, J. Int. Seqs. Vol. 12 (2009) 09.3.3.

FORMULA

a(n,m) = sum(product(S1(-1;j,1)^e(n,m,q,j),j=1..n),q=1..p(n,m)) if n>=m>=1, else 0. Here p(n,m)=A008284(n,m), the number of m parts partitions of n and e(n,m,q,j) is the exponent of j in the q-th m part partition of n. S1(-1,n,1)= A008279(1,n-1) = [1,1,0,0,0,...], n>=1.

THe triangle starts in row n with ceiling(n/2) - 1 zeros, and is 1 otherwise. - Wolfdieter Lang, Aug 03 2023

EXAMPLE

Triangle begins:

[1];

[1,1];

[0,1,1];

[0,1,1,1];

[0,0,1,1,1];

[0,0,1,1,1,1];

...

CROSSREFS

Cf. A004526(n+2), n>=1, (row sums).