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

Author

Darío Clavijo, May 26 2025

Keywords

• nonn

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

• Mancala World, Tchoukaillon

Crossrefs

Cf. A000707, A008302.

Programs

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 *)

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)])

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).