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%I #21 Feb 28 2020 22:53:18
%S 1,0,1,2,2,4,3,9,15,27,40,52,88,148,256,205,405,665,1105,1845,3125,
%T 2556,3786,6216,10206,16836,27906,46656,24409,42301,68803,112315,
%U 183757,301609,496951,823543,347712,542984,881392,1431816,2330336,3800392,6213264,10188872,16777216
%N Triangle read by rows: T(n,k) (1 <= k <= n) defined by T(n,n) = (n-1)^(n-1), T(n,k) = T(n,k+1) - (n-1)*T(n-1,k) for k = n-1 .. 1.
%C Arises in analysis of game with n players: each person picks a number from 1 to n, and the winner is the largest unique choice (see Guy's letter). T(n,k) is the number out of all possible games (i.e., all n^n sets of choices) which are won by a given player who has chosen k.
%H R. K. Guy, <a href="/A241580/a241580_1.pdf">Letter to N. J. A. Sloane, Jun 21, 1975</a>
%e Triangle begins:
%e 1;
%e 0, 1;
%e 2, 2, 4;
%e 3, 9, 15, 27;
%e 40, 52, 88, 148, 256;
%e 205, 405, 665, 1105, 1845, 3125;
%e 2556, 3786, 6216, 10206, 16836, 27906, 46656;
%e 24409, 42301, 68803, 112315, 183757, 301609, 496951, 823543;
%e ...
%p M:=20;
%p M2:=10;
%p T[1,1]:=1:
%p for n from 2 to M do
%p T[n,n]:=(n-1)^(n-1);
%p for k from n-1 by -1 to 1 do
%p T[n,k]:=T[n,k+1]-(n-1)*T[n-1,k]:
%p od:
%p od:
%p for n from 1 to M2 do lprint([seq(T[n,k],k=1..n)]); od:
%Y T(n,0) is A231797, row sums are A241581.
%K nonn,tabl
%O 1,4
%A _N. J. A. Sloane_, Apr 29 2014
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