A381891 Triangle read by rows: T(n,k) is the number of partitions of a 2-colored set of n objects into at most k parts with 0 <= k <= n.

1, 0, 2, 0, 3, 6, 0, 4, 10, 14, 0, 5, 19, 28, 33, 0, 6, 28, 52, 64, 70, 0, 7, 44, 93, 127, 142, 149, 0, 8, 60, 152, 228, 272, 290, 298, 0, 9, 85, 242, 404, 507, 561, 582, 591, 0, 10, 110, 370, 672, 904, 1034, 1098, 1122, 1132, 0, 11, 146, 546, 1100, 1568, 1870, 2027, 2101, 2128, 2139

Offset

0,3

Comments

The 1-color case is Euler's table A026820.

Links

Alois P. Heinz, Rows n = 0..150, flattened

Formula

T(1,k) = k + 1.

T(n,n) = A005380(n).

Example

Triangle begins:
  1;
  0, 2;
  0, 3,  6;
  0, 4, 10,  14;
  0, 5, 19,  28,  33;
  0, 6, 28,  52,  64,  70;
  0, 7, 44,  93, 127, 142, 149;
  0, 8, 60, 152, 228, 272, 290, 298;
  ...

Maple

b:= proc(n, i) option remember; expand(`if`(n=0 or i=1, (n+1)*x^n,
      add(b(n-i*j, min(n-i*j, i-1))*binomial(i+j, j)*x^j, j=0..n/i)))
    end:
T:= proc(n, k) option remember;
      `if`(k<0, 0, T(n, k-1)+coeff(b(n$2), x, k))
    end:
seq(seq(T(n, k), k=0..n), n=0..10);  # Alois P. Heinz, Mar 09 2025

Mathematica

b[n_, i_] := b[n, i] = Expand[If[n == 0 || i == 1, (n+1)*x^n, Sum[b[n-i*j, Min[n-i*j, i-1]]*Binomial[i+j, j]*x^j, {j, 0, n/i}]]];
T[n_, k_] := T[n, k] = If[k < 0, 0, T[n, k-1] + Coefficient[b[n, n], x, k]];
Table[Table[T[n, k], {k, 0, n}], {n, 0, 10}] // Flatten (* Jean-François Alcover, Nov 24 2025, after Alois P. Heinz *)

Prog

(Python)
from sympy import binomial
from sympy.utilities.iterables import partitions
from sympy.combinatorics.partitions import IntegerPartition
def a381891_row( n):
    if n == 0 : return [1]
    t = list( [0] * n)
    for p in partitions( n):
        p = IntegerPartition( p).as_dict()
        fact = 1
        s = 0
        for k in p :
            s += p[k]
            fact *= binomial( k + p[k], p[k])
        if s > 0 :
            t[s - 1] += fact
    for i in range( n - 1):
        t[i+1] += t[i]
    return [0] + t

Crossrefs

Cf. A005380, A026820.

Sequence in context: A155800 A276658 A079510 * A386789 A216255 A362788

Adjacent sequences: A381888 A381889 A381890 * A381892 A381893 A381894

Keyword

nonn,tabl

Author

Peter Dolland, Mar 09 2025

Status

approved