A382522 Square array A(n,k), n>=0, k>=0, read by antidiagonals downwards, where n unlabeled objects are distributed into k containers of four kinds. Containers may be left empty.

1, 4, 0, 10, 4, 0, 20, 16, 4, 0, 35, 40, 26, 4, 0, 56, 80, 80, 32, 4, 0, 84, 140, 180, 124, 42, 4, 0, 120, 224, 340, 320, 184, 48, 4, 0, 165, 336, 574, 660, 535, 248, 58, 4, 0, 220, 480, 896, 1184, 1200, 800, 332, 64, 4, 0, 286, 660, 1320, 1932, 2284, 1956, 1176, 416, 74, 4, 0

Offset

0,2

Links

Table of n, a(n) for n=0..65.

Formula

A(0,k) = binomial(k + 3, 3) = A000292(k + 1).

A(1,k) = 4 * binomial(k + 2, 3).

A(n,1) = 4.

A(n,k) = Sum_{i=0..k} binomial(k + 3 - i, 3) * A382344(n,i) for k <= n.

A(n,k) = A382344(n+k,k).

Example

Array starts:
 0 : [1, 4, 10,  20,   35,    56,    84,   120,    165,    220,    286]
 1 : [0, 4, 16,  40,   80,   140,   224,   336,    480,    660,    880]
 2 : [0, 4, 26,  80,  180,   340,   574,   896,   1320,   1860,   2530]
 3 : [0, 4, 32, 124,  320,   660,  1184,  1932,   2944,   4260,   5920]
 4 : [0, 4, 42, 184,  535,  1200,  2284,  3892,   6129,   9100,  12910]
 5 : [0, 4, 48, 248,  800,  1956,  3968,  7088,  11568,  17660,  25616]
 6 : [0, 4, 58, 332, 1176,  3080,  6618, 12364,  20892,  32776,  48590]
 7 : [0, 4, 64, 416, 1616,  4560, 10368, 20280,  35536,  57376,  87040]
 8 : [0, 4, 74, 520, 2187,  6580, 15778, 32196,  58414,  97012, 150570]
 9 : [0, 4, 80, 628, 2848,  9140, 23088, 49172,  92352, 157808, 250720]
10 : [0, 4, 90, 752, 3660, 12440, 33002, 73188, 142160, 249740, 406036]
...

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))*binomial(j+3, 3), 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..10);  # Alois P. Heinz, Mar 31 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]]*Binomial[j+3, 3], {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, 10}] // Flatten (* Jean-François Alcover, Aug 07 2025, after Alois P. Heinz *)

Prog

(Python)
from sympy import binomial
from sympy.utilities.iterables import partitions
def a_row(n, length=11) :
    if n == 0 : return [ binomial( k + 3, 3) for k in range( length) ]
    t = list( [0] * length)
    for p in partitions( n):
        fact = 1
        s = 0
        for k in p :
            s += p[k]
            fact *= binomial( 3 + p[k], 3)
        if s > 0 :
            t[s] += fact
    a = list( [0] * length)
    for i in range( 1, length):
        for j in range( i, 0, -1):
            a[i] += t[j] * binomial( i - j + 3, 3)
    return a
for n in range(11): print(a_row(n))

Crossrefs

Antidiagonal sums give A023003.

Without empty containers: A382041.

Cf. A382344, A000292, 2 kinds: A382345, 3 kinds: A382521.

Sequence in context: A390040 A164735 A390039 * A293933 A345057 A158976

Adjacent sequences: A382519 A382520 A382521 * A382523 A382524 A382525

Keyword

nonn,tabl

Author

Peter Dolland, Mar 31 2025

Status

approved