A382454 Number of solutions winning the Tchoukaillon game with n seeds and 2n pits.

1, 2, 9, 49, 285, 1717, 10569, 66013, 416687, 2651355, 16976806, 109256095, 706071989, 4579020513, 29784426945, 194231327451, 1269457354069, 8313189986612, 54534379879411, 358298017624625, 2357331709694072, 15528887031395023, 102412421113465576, 676104332189192702

Offset

0,2

Comments

a(n) is the number of permutations of [2n+1] with n inversions. a(2) = 9: 12453, 12534, 13254, 13425, 14235, 21354, 21435, 23145, 31245. - Alois P. Heinz, May 27 2025

Links

Table of n, a(n) for n=0..23.

Mancala World, Tchoukaillon

Formula

a(n) = T(2n,n) with T(x,y) = Sum_{v=0..min(x,y)} T(x-1, y-v) and T(0,y) = 1 if y = 0 else 0.

a(n) = A008302(2n+1,n).

Maple

a:= n-> coeff(series(mul((1-q^j)/(1-q), j=1..2*n+1), q, n+1), q, n):
seq(a(n), n=0..23);  # Alois P. Heinz, May 27 2025

Mathematica

a[n_]:=Coefficient[Series[Product[(1-q^j)/(1-q), {j, 1, 2*n+1}], {q, 0, n+1}]//Normal, q, n]; Array[a, 24, 0] (* Shenghui Yang, Jun 02 2025 *)

Prog

(Python)
def a(n):
    if n == 0: return 1
    p = [1]
    for j in range(1, (n << 1) + 2):
        np = [0] * (len(p) + j - 1)
        for k in range(len(p)):
            for l in range(j):
                if (kl:=k+l) <= n:
                    np[kl] += p[k]
        p = np[:n+1]
    return p[n]
print([a(n) for n in range(1, 24)])

Crossrefs

Cf. A000707, A008302.

Sequence in context: A375798 A224140 A360103 * A390570 A389435 A370345

Adjacent sequences: A382451 A382452 A382453 * A382455 A382456 A382457

Keyword

nonn

Author

Darío Clavijo, May 26 2025

Status

approved