A339884 -id:A339884 - 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

A232539 Triangle read by rows: T(n,k) = number of partitions of n into at most four parts in which the largest part is equal to k, 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 0, 0, 2, 2, 1, 1, 0, 0, 2, 3, 2, 1, 1, 0, 0, 1, 3, 3, 2, 1, 1, 0, 0, 1, 3, 4, 3, 2, 1, 1, 0, 0, 0, 3, 4, 4, 3, 2, 1, 1, 0, 0, 0, 2, 5, 5, 4, 3, 2, 1, 1, 0, 0, 0, 1, 4, 6, 5, 4, 3, 2, 1, 1, 0, 0, 0, 1, 4, 6, 7, 5, 4, 3, 2, 1, 1

Offset: 0

L. Edson Jeffery, Jan 02 2014

Also number of partitions of n into k parts with parts in the range 1..4.

			Triangle T{n,k} begins:
  1;
  0, 1;
  0, 1, 1;
  0, 1, 1, 1;
  0, 1, 2, 1, 1;
  0, 0, 2, 2, 1, 1;
  0, 0, 2, 3, 2, 1, 1;
  0, 0, 1, 3, 3, 2, 1, 1;
  0, 0, 1, 3, 4, 3, 2, 1, 1;
  0, 0, 0, 3, 4, 4, 3, 2, 1, 1;
  ...

Louis Comtet, Advanced Combinatorics, Reidel (1974).

Cf. A001400 (row sums), A219237, A233292 (row partial sums), A145362 (parts <=2), A339884 (parts <=3).

maxp := 4 :
gf := 1/mul(1-u*t^i,i=1..maxp) :
for n from 0 to 13 do
    for m from 0 to n do
        coeftayl(gf,t=0,n) ;
        coeftayl(%,u=0,m) ;
        printf("%d ",%);
    end do:
    printf("\n") ;
end do: # R. J. Mathar, May 27 2025

G.f.: 1/((1-u*t)*(1-u*t^2)*(1-u*t^3)*(1-u*t^4)). - [Comtet p. 97 [2c]]. - R. J. Mathar, May 27 2025

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

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