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%I #36 Dec 20 2022 11:29:32
%S 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 108,94,63,127,190,220,232,232,220,190,127,255,382,444,472,480,472,
%U 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 Triangle read by rows: T(n, k) = 2^n - 2^(n-k-1) - 2^k, 0 <= k <= n-1.
%C 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 T(n, k) = 2^n - 2^(n-k-1) - 2^k.
%F Sum_{k=0..n-1} T(n,k)*binomial(n-1,k) = 2*A005061(n-1)
%e Triangle begins:
%e 0;
%e 1, 1;
%e 3, 4, 3;
%e 7, 10, 10, 7;
%e 15, 22, 24, 22, 15;
%e 31, 46, 52, 52, 46, 31;
%e 63, 94, 108, 112, 108, 94, 63;
%e 127, 190, 220, 232, 232, 220, 190, 127;
%e 255, 382, 444, 472, 480, 472, 444, 382, 255;
%e 511, 766, 892, 952, 976, 976, 952, 892, 766, 511;
%e 1023, 1534, 1788, 1912, 1968, 1984, 1968, 1912, 1788, 1534, 1023;
%e 2047, 3070, 3580, 3832, 3952, 4000, 4000, 3952, 3832, 3580, 3070, 2047;
%e ...
%p T := n -> seq(2^n - 2^(n - k - 1) - 2^k, k = 0 .. n - 1);
%p seq(T(n), n=1..12);
%t 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 Column k=0 gives A000225(n-1).
%Y Column k=1 gives A033484(n-2).
%Y Column k=2 gives A053208(n-3).
%Y Cf. A005061, A359200.
%K nonn,easy,tabl
%O 1,4
%A _Ambrosio Valencia-Romero_, Dec 20 2022
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