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

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, 108, 94, 63, 127, 190, 220, 232, 232, 220, 190, 127, 255, 382, 444, 472, 480, 472, 444, 382, 255, 511, 766, 892, 952, 976, 976, 952, 892, 766, 511, 1023, 1534, 1788, 1912, 1968, 1984, 1968, 1912, 1788, 1534, 1023

Offset

1,4

Comments

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.

Links

Table of n, a(n) for n=1..66.

Formula

T(n, k) = 2^n - 2^(n-k-1) - 2^k.

Sum_{k=0..n-1} T(n,k)*binomial(n-1,k) = 2*A005061(n-1)

Example

Triangle begins:
  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,  108,   94,   63;
 127,  190,  220,  232,  232,  220,  190,  127;
 255,  382,  444,  472,  480,  472,  444,  382,  255;
 511,  766,  892,  952,  976,  976,  952,  892,  766,  511;
1023, 1534, 1788, 1912, 1968, 1984, 1968, 1912, 1788, 1534, 1023;
2047, 3070, 3580, 3832, 3952, 4000, 4000, 3952, 3832, 3580, 3070, 2047;
  ...

Maple

T := n -> seq(2^n - 2^(n - k - 1) - 2^k, k = 0 .. n - 1);
seq(T(n), n=1..12);

Mathematica

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 *)

Crossrefs

Column k=0 gives A000225(n-1).

Column k=1 gives A033484(n-2).

Column k=2 gives A053208(n-3).

Cf. A005061, A359200.

Sequence in context: A291317 A141730 A127737 * A299416 A333974 A163108

Adjacent sequences: A358122 A358123 A358124 * A358126 A358127 A358128

Keyword

nonn,easy,tabl

Author

Ambrosio Valencia-Romero, Dec 20 2022

Status

approved