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A358125
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Triangle read by rows: T(n, k) = 2^n - 2^(n-k-1) - 2^k, 0 <= k <= n-1.
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2
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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
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OFFSET
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1,4
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COMMENTS
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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.
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LINKS
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FORMULA
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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)
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EXAMPLE
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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;
...
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MAPLE
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T := n -> seq(2^n - 2^(n - k - 1) - 2^k, k = 0 .. n - 1);
seq(T(n), n=1..12);
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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