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A350770
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Triangle read by rows: T(n, k) = 2^(n-k-1) + 2^k - 2, 0 <= k <= n-1.
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2
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0, 1, 1, 3, 2, 3, 7, 4, 4, 7, 15, 8, 6, 8, 15, 31, 16, 10, 10, 16, 31, 63, 32, 18, 14, 18, 32, 63, 127, 64, 34, 22, 22, 34, 64, 127, 255, 128, 66, 38, 30, 38, 66, 128, 255, 511, 256, 130, 70, 46, 46, 70, 130, 256, 511, 1023, 512, 258, 134, 78, 62, 78, 134, 258, 512, 1023, 2047, 1024, 514, 262, 142, 94, 94, 142, 262, 514, 1024, 2047
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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 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 without 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.
The sum of the products of T(n, k) and binomial(n-1,k) for 0 <= k <= n-1 equals 2*A001047(n-1). For instance, for n = 3, T(3, k) returns 3, 2, and 3 and binomial(3-1,k) returns 1, 2, and 1 for k = 0, 1, and 2, respectively. Then 3*1 + 2*2 + 3*1 = 2*A001047(3-1) = 2*5 = 10. Similarly, for n = 4, the result yields 7*1 + 4*3 + 4*3 + 7*1 = 2*A001047(4-1) = 2*19 = 38.
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LINKS
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FORMULA
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T(n, k) = 2^(n-k-1) + 2^k - 2.
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EXAMPLE
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Triangle begins:
0;
1, 1;
3, 2, 3;
7, 4, 4, 7;
15, 8, 6, 8, 15;
31, 16, 10, 10, 16, 31;
63, 32, 18, 14, 18, 32, 63;
127, 64, 34, 22, 22, 34, 64, 127;
255, 128, 66, 38, 30, 38, 66, 128, 255;
511, 256, 130, 70, 46, 46, 70, 130, 256, 511;
1023, 512, 258, 134, 78, 62, 78, 134, 258, 512, 1023;
2047, 1024, 514, 262, 142, 94, 94, 142, 262, 514, 1024, 2047;
...
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MAPLE
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T := n -> seq(2^(n - k - 1) + 2^k - 2, k = 0 .. n - 1);
seq(T(n), n=1..12);
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PROG
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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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