A145361 -id:A145361 - cp's OEIS frontend

A145362 Lower triangular array S1hat(-1) read by rows, 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

Offset: 1

Wolfdieter Lang, Oct 17 2008

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

			Triangle begins:
  [1];
  [1,1];
  [0,1,1];
  [0,1,1,1];
  [0,0,1,1,1];
  [0,0,1,1,1,1];
  ...

Wolfdieter Lang, First 10 rows of the array and more.
Louis Comtet, Advanced Combinatorics, Reidel (1974).
Wolfdieter Lang, Combinatorial Interpretation of Generalized Stirling Numbers, J. Int. Seqs. Vol. 12 (2009) 09.3.3.

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

Cf. A008275, A008279, A008284, A036039, A145361, A339884 (parts <=3), A232539 (parts <=4).

T(n, k) = n<=2*k; \\ Jinyuan Wang, Jan 19 2025

T(n,m) = Sum_{q=1..p(n,m)} Product_{j=1..n} S1(-1;j,1)^e(n,m,q,j) 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
G.f.: 1/((1-u*t)*(1-u*t^2)). [Comtet page 97 [2c]]. - R. J. Mathar, May 27 2025

A145363 Partition number array, called M31hat(-2).

Original entry on oeis.org

1, 2, 1, 2, 2, 1, 0, 2, 4, 2, 1, 0, 0, 4, 2, 4, 2, 1, 0, 0, 0, 4, 0, 4, 8, 2, 4, 2, 1, 0, 0, 0, 0, 0, 0, 4, 8, 0, 4, 8, 2, 4, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 4, 8, 16, 0, 4, 8, 2, 4, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 8, 16, 0, 0, 4, 8, 16, 0, 4, 8, 2, 4, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Wolfdieter Lang, Oct 17 2008

If all positive numbers are replaced by 1 this becomes the characteristic partition array for partitions with parts 1,2 or 3 only, provided the partitions of n are ordered like in Abramowitz-Stegun (A-St order; for the reference see A134278).
Second member (K=2) in the family M31hat(-K) of partition number arrays.
The sequence of row lengths is A000041 (partition numbers) [1, 2, 3, 5, 7, 11, 15, 22, 30, 42,...].
This array is array A144358 divided entrywise by the array M_3=M3(1)=A036040. Formally 'A144358/A036040'. E.g. a(4,3)= 4 = 12/3 = A144358(4,3)/A036040(4,3).
If M31hat(-2;n,k) is summed over those k belonging to partitions with fixed number of parts m one obtains the unsigned triangle S1hat(-2):= A145364.

			Triangle begins
  [1];
  [2,1];
  [2,2,1];
  [0,2,4,2,1];
  [0,0,4,2,4,2,1];
  ...
a(4,3)= 4 = S1(-2;2,1)^2. The relevant partition of 4 is (2^2).

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.

Cf. A145361 (M31hat(-1)). A145366 (M31hat(-3)).

a(n,k) = product(S1(-2;j,1)^e(n,k,j),j=1..n) with S1(-2;n,1) = A008279(2,n-1) = [1,2,1,0,0,0,...], n>=1 and the exponent e(n,k,j) of j in the k-th partition of n in the A-St ordering of the partitions of n.

cp's OEIS Frontend

A145362 Lower triangular array S1hat(-1) read by rows, related to partition number array A145361.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula