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A372967 Number of compositions of 2n into 2n nonnegative parts such that their xor-sum is zero.

%I A372967 #25 Apr 02 2025 04:12:36
%S A372967 1,1,7,226,2059,20926,315646,4397212,66201971,999067510,15168583482,
%T A372967 240202475668,3731763898510,57290627029676,887861389544668,
%U A372967 13713341876387776,210889953761225667,3248614469788303782,50091681144815341810,772966100038376636332
%N A372967 Number of compositions of 2n into 2n nonnegative parts such that their xor-sum is zero.
%C A372967 Number of starting configurations of Nim with 2n pieces such that 2nd player wins, and the configurations are of the form {x_1, x_2, ..., x_2n}, where x_i is the number of pieces on i-th stack (x_i>=0), and the sum of all pieces is 2n.
%H A372967 C. L. Bouton, <a href="https://www.jstor.org/stable/1967631">Nim, A Game with a Complete Mathematical Theory</a>, Annals of Mathematics, Second Series, vol. 3 (1/4), 1902, 35-39.
%F A372967 a(n) = A088218(2n) - A372871(2n).
%F A372967 a(n) mod 2 = 1 <=> n in { A131577 }.
%e A372967 For n=1 the a(1)=1 solution is {1,1}.
%e A372967 For n=2 the a(2)=7 solutions are {0,0,2,2}, {0,2,0,2}, {0,2,2,0}, {1,1,1,1}, {2,0,0,2}, {2,0,2,0}, {2,2,0,0}.
%p A372967 b:= proc(n, i, t) option remember; `if`(n=0, 1-signum(t),
%p A372967       add(b(n-j, i-1, Bits[Xor](j, t)), j=`if`(i=1, n, 0..n)))
%p A372967     end:
%p A372967 a:= n-> b(2*n$2, 0):
%p A372967 seq(a(n), n=0..23);  # _Alois P. Heinz_, May 22 2024
%t A372967 b[n_, i_, t_] := b[n, i, t] = If[n == 0, 1-Sign[t],
%t A372967    Sum[b[n-j, i-1, BitXor[j, t]], {j, If[i == 1, {n}, Range[0, n]]}]];
%t A372967 a[n_] := b[2n, 2n, 0];
%t A372967 Table[a[n], {n, 0, 23}] (* _Jean-François Alcover_, May 30 2024, after _Alois P. Heinz_ *)
%Y A372967 Cf. A088218, A131577, A372871, A048833, A233810.
%K A372967 nonn
%O A372967 0,3
%A A372967 _Anna Ledworowska_, May 18 2024
%E A372967 More terms from _Alois P. Heinz_, May 22 2024