A382345 Square array A(n,k), n>=0, k>=0, read by antidiagonals downwards, where n unlabeled objects are distributed into k containers of two kinds. Containers may be left empty.

1, 2, 0, 3, 2, 0, 4, 4, 2, 0, 5, 6, 7, 2, 0, 6, 8, 12, 8, 2, 0, 7, 10, 17, 18, 11, 2, 0, 8, 12, 22, 28, 26, 12, 2, 0, 9, 14, 27, 38, 46, 34, 15, 2, 0, 10, 16, 32, 48, 66, 64, 46, 16, 2, 0, 11, 18, 37, 58, 86, 100, 94, 56, 19, 2, 0, 12, 20, 42, 68, 106, 136, 152, 124, 70, 20, 2, 0

Offset: 0

Author

Peter Dolland, Mar 29 2025

Keywords

• nonn
• tabl

Examples

			Array starts:
 0 : [1, 2,  3,   4,   5,   6,   7,    8,    9,   10,   11]
 1 : [0, 2,  4,   6,   8,  10,  12,   14,   16,   18,   20]
 2 : [0, 2,  7,  12,  17,  22,  27,   32,   37,   42,   47]
 3 : [0, 2,  8,  18,  28,  38,  48,   58,   68,   78,   88]
 4 : [0, 2, 11,  26,  46,  66,  86,  106,  126,  146,  166]
 5 : [0, 2, 12,  34,  64, 100, 136,  172,  208,  244,  280]
 6 : [0, 2, 15,  46,  94, 152, 217,  282,  347,  412,  477]
 7 : [0, 2, 16,  56, 124, 214, 316,  426,  536,  646,  756]
 8 : [0, 2, 19,  70, 167, 302, 464,  640,  825, 1010, 1195]
 9 : [0, 2, 20,  84, 212, 406, 648,  922, 1212, 1512, 1812]
10 : [0, 2, 23, 100, 271, 542, 899, 1314, 1766, 2236, 2717]
...
		

Links

• Alois P. Heinz, Antidiagonals n = 0..200, flattened

Crossrefs

Antidiagonal sums give A000712.

Alternating antidiagonal sums give A073252.

Without empty containers: A381895.

Cf. A382342.

Programs

Maple

b:= proc(n, i) option remember; expand(`if`(n=0, 1, `if`(i<1, 0,
      add(x^j*b(n-i*j, min(n-i*j, i-1))*(j+1), j=0..n/i))))
    end:
A:= (n, k)-> coeff(b(n+k$2), x, k):
seq(seq(A(n, d-n), n=0..d), d=0..11);  # Alois P. Heinz, Mar 29 2025

Mathematica

b[n_, i_] := b[n, i] = Expand[If[n == 0, 1, If[i < 1, 0,
   Sum[x^j*b[n - i*j, Min[n - i*j, i - 1]]*(j + 1), {j, 0, n/i}]]]];
A[n_, k_] := Coefficient[b[n + k, n + k], x, k];
Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 11}] // Flatten (* Jean-François Alcover, Apr 07 2025, after Alois P. Heinz *)

Python

from sympy.utilities.iterables import partitions
def a_row(n, length=11) -> list[int]:
    if n == 0 : return list(range(1, length + 1))
    t = [0] * length
    for p in partitions(n):
        fact = 1
        s = 0
        for k in p :
            s += p[k]
            fact *= p[k] + 1
        if s > 0 :
            t[s] += fact
    for i in range(1, length - 1):
        t[i+1] += t[i] * 2 - t[i-1]
    return t
for n in range(11): print(a_row(n))

Formula

A(0,k) = k + 1.

A(1,k) = 2*k.

A(2,k+1) = 2 + 5 * k.

A(n,1) = 2.

A(n,k) = Sum_{i=0..k} (k + 1 - i) * A382342(n,i) for k <= n.

A(n,n+k) = A(n,n) + k * A000712(n).

A(n,k) = A382342(n,k) + 2 * A(n,k-1) - A(n,k-2) for 2 <= k <= n.

A(n,k) = A382342(n+k,k). - Alois P. Heinz, Mar 31 2025