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A358125 Triangle read by rows: T(n, k) = 2^n - 2^(n-k-1) - 2^k, 0 <= k <= n-1.

%I A358125 #36 Dec 20 2022 11:29:32
%S A358125 0,1,1,3,4,3,7,10,10,7,15,22,24,22,15,31,46,52,52,46,31,63,94,108,112,
%T A358125 108,94,63,127,190,220,232,232,220,190,127,255,382,444,472,480,472,
%U A358125 444,382,255,511,766,892,952,976,976,952,892,766,511,1023,1534,1788,1912,1968,1984,1968,1912,1788,1534,1023
%N A358125 Triangle read by rows: T(n, k) = 2^n - 2^(n-k-1) - 2^k, 0 <= k <= n-1.
%C A358125 T(n, k) is the expanded number of player-reduced static games within an n-player two-strategy game scenario in which one player (the "standpoint") faces a specific combination of other players' individual strategies with the possibility of anti-coordination between them -- the total number of such combinations is 2^(n-1). The value of k represents the number of other players who (are expected to) agree on one of the two strategies in S, while the other n-k-1 choose the other strategy; the standpoint player is not included.
%F A358125 T(n, k) = 2^n - 2^(n-k-1) - 2^k.
%F A358125 Sum_{k=0..n-1} T(n,k)*binomial(n-1,k) = 2*A005061(n-1)
%e A358125 Triangle begins:
%e A358125   0;
%e A358125   1,     1;
%e A358125   3,     4,    3;
%e A358125   7,    10,   10,    7;
%e A358125   15,   22,   24,   22,   15;
%e A358125   31,   46,   52,   52,   46,   31;
%e A358125   63,   94,  108,  112,  108,   94,   63;
%e A358125  127,  190,  220,  232,  232,  220,  190,  127;
%e A358125  255,  382,  444,  472,  480,  472,  444,  382,  255;
%e A358125  511,  766,  892,  952,  976,  976,  952,  892,  766,  511;
%e A358125 1023, 1534, 1788, 1912, 1968, 1984, 1968, 1912, 1788, 1534, 1023;
%e A358125 2047, 3070, 3580, 3832, 3952, 4000, 4000, 3952, 3832, 3580, 3070, 2047;
%e A358125   ...
%p A358125 T := n -> seq(2^n - 2^(n - k - 1) - 2^k, k = 0 .. n - 1);
%p A358125 seq(T(n), n=1..12);
%t A358125 T[n_, k_] := 2^n - 2^(n - k - 1) - 2^k; Table[T[n, k], {n, 1, 11}, {k, 0, n - 1}] // Flatten (* _Amiram Eldar_, Dec 20 2022 *)
%Y A358125 Column k=0 gives A000225(n-1).
%Y A358125 Column k=1 gives A033484(n-2).
%Y A358125 Column k=2 gives A053208(n-3).
%Y A358125 Cf. A005061, A359200.
%K A358125 nonn,easy,tabl
%O A358125 1,4
%A A358125 _Ambrosio Valencia-Romero_, Dec 20 2022