A358125 - OEIS

%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