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

%I A382454 #40 Jun 02 2025 18:28:06
%S A382454 1,2,9,49,285,1717,10569,66013,416687,2651355,16976806,109256095,
%T A382454 706071989,4579020513,29784426945,194231327451,1269457354069,
%U A382454 8313189986612,54534379879411,358298017624625,2357331709694072,15528887031395023,102412421113465576,676104332189192702
%N A382454 Number of solutions winning the Tchoukaillon game with n seeds and 2n pits.
%C A382454 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
%H A382454 Mancala World, <a href="https://mancala.fandom.com/wiki/Tchoukaillon">Tchoukaillon</a>
%F A382454 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.
%F A382454 a(n) = A008302(2n+1,n).
%p A382454 a:= n-> coeff(series(mul((1-q^j)/(1-q), j=1..2*n+1), q, n+1), q, n):
%p A382454 seq(a(n), n=0..23);  # _Alois P. Heinz_, May 27 2025
%t A382454 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 *)
%o A382454 (Python)
%o A382454 def a(n):
%o A382454     if n == 0: return 1
%o A382454     p = [1]
%o A382454     for j in range(1, (n << 1) + 2):
%o A382454         np = [0] * (len(p) + j - 1)
%o A382454         for k in range(len(p)):
%o A382454             for l in range(j):
%o A382454                 if (kl:=k+l) <= n:
%o A382454                     np[kl] += p[k]
%o A382454         p = np[:n+1]
%o A382454     return p[n]
%o A382454 print([a(n) for n in range(1,24)])
%Y A382454 Cf. A000707, A008302.
%K A382454 nonn
%O A382454 0,2
%A A382454 _Darío Clavijo_, May 26 2025