A145362 -id:A145362 - cp's OEIS frontend

A155091 Triangle read by rows. Signed version of A145362. Main diagonal positive, rest of the nonzero terms negative.

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

Offset: 1

Mats Granvik, Jan 20 2009

Matrix inverse of this triangle is A155092.

			Table begins:
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,

Cf. A145362.

A145361 Characteristic partition array for partitions with parts 1 and 2 only.

Original entry on oeis.org

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

Offset: 1

Wolfdieter Lang, Oct 17 2008

Each partition of n, ordered like in Abramowitz-Stegun (A-St order; for the reference see A134278), is mapped to 1 if the partition has parts 1 or 2 only and to 0 otherwise.
First member (K=1) 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 A144357 divided entrywise by the array M_3=M3(1)=A036040. Formally 'A144357/A036040'. E.g. a(4,3)= 1 = 3/3 = A144357(4,3)/A036040(4,3).
If M31hat(-1;n,k) is summed over those k numerating partitions with fixed number of parts m one obtains the unsigned triangle S1hat(-1):= A145362 .

			Triangle begins:
  [1];
  [1,1];
  [0,1,1];
  [0,0,1,1,1];
  [0,0,0,0,1,1,1];
  ...
a(4,3)= 1 = S1(-1;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. A145363 (M31hat(-2)).

a(n,k) = product(S1(-1;j,1)^e(n,k,j),j=1..n) with S1(-1;n,1) = A008279(1,n-1) = [1,1,0,0,0,...], 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.

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

A339884 Triangle read by rows: T(n, m) gives the number of partitions of n with m parts and parts from {1, 2, 3}.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 0, 2, 1, 1, 0, 1, 2, 1, 1, 0, 1, 2, 2, 1, 1, 0, 0, 2, 2, 2, 1, 1, 0, 0, 1, 3, 2, 2, 1, 1, 0, 0, 1, 2, 3, 2, 2, 1, 1, 0, 0, 0, 2, 3, 3, 2, 2, 1, 1, 0, 0, 0, 1, 3, 3, 3, 2, 2, 1, 1, 0, 0, 0, 1, 2, 4, 3, 3, 2, 2, 1, 1

Offset: 1

Wolfdieter Lang, Jan 31 2021

Row sums give A001399(n), for n >= 1.

One could add the column [1,repeat 0] for m = 0 starting with n >= 0.

			The triangle T(n,m) begins:
  n\m  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 ...
  1:   1
  2:   1 1
  3:   1 1 1
  4:   0 2 1 1
  5:   0 1 2 1 1
  6:   0 1 2 2 1 1
  7:   0 0 2 2 2 1 1
  8:   0 0 1 3 2 2 1 1
  9:   0 0 1 2 3 2 2 1 1
  10:  0 0 0 2 3 3 2 2 1  1
  11:  0 0 0 1 3 3 3 2 2  1  1
  12:  0 0 0 1 2 4 3 3 2  2  1  1
  13:  0 0 0 0 2 3 4 3 3  2  2  1  1
  14:  0 0 0 0 1 3 4 4 3  3  2  2  1  1
  15:  0 0 0 0 1 2 4 4 4  3  3  2  2  1  1
  16:  0 0 0 0 0 2 3 5 4  4  3  3  2  2  1  1
  17:  0 0 0 0 0 1 3 4 5  4  4  3  3  2  2  1  1
  18:  0 0 0 0 0 1 2 4 5  5  4  4  3  3  2  2  1  1
  19:  0 0 0 0 0 0 2 3 5  5  5  4  4  3  3  2  2  1  1
  20:  0 0 0 0 0 0 1 3 4  6  5  5  4  4  3  3  2  2  1  1
  ...
Row n = 6: the partitions of 6 with number of parts m = 1,2, ...., 6, and parts from {1,2,3} are (in Abramowitz-Stegun order): [] | [],[],[3,3] | [],[1,2,3],[2^3] | [1^3,3],[1^2,2^2] | [1^4,2] | 1^6, giving 0, 1, 2, 2, 1, 1.

Louis Comtet, Advanced Combinatorics, Reidel (1974)

Cf. A001399, A008284 (all parts), A145362 (parts 1, 2), A232539 (parts <=4), A291983.

Compositions: A007818, A030528 (parts 1, 2), A078803 (parts 1, 2, 3).

Sum_{k=0..n} (-1)^k * T(n,k) = A291983(n). - Alois P. Heinz, Feb 01 2021

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

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A155091 Triangle read by rows. Signed version of A145362. Main diagonal positive, rest of the nonzero terms negative.

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A145361 Characteristic partition array for partitions with parts 1 and 2 only.

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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.

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A339884 Triangle read by rows: T(n, m) gives the number of partitions of n with m parts and parts from {1, 2, 3}.

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