%I #9 Mar 31 2012 10:25:33

%S 1,1,1,1,2,2,6,1,3,4,19,3,10,16,4,1,2,7,37,6,31,48,4,4,14,24,5,26,152,

%T 12,80,1,2,4,51,12,39,100,20,8,23,90,4,81,14,8,242,5,12,36,4,38,215,

%U 16,172,36,190,395,40,24

%N Number of evaluation schemes for x^n achieving the minimal number of multiplications.

%e For n=7, we can evaluate x^7 using only 4 multiplications in 6 ways:

%e x^2 = x * x ; x^3 = x * x^2 ; x^4 = x * x^3 ; x^7 = x^3 * x^4

%e x^2 = x * x ; x^3 = x * x^2 ; x^4 = x^2 * x^2 ; x^7 = x^3 * x^4

%e x^2 = x * x ; x^3 = x * x^2 ; x^5 = x^2 * x^3 ; x^7 = x^2 * x^5

%e x^2 = x * x ; x^3 = x * x^2 ; x^6 = x^3 * x^3 ; x^7 = x * x^6

%e x^2 = x * x ; x^4 = x^2 * x^2 ; x^5 = x * x^4 ; x^7 = x^2 * x^5

%e x^2 = x * x ; x^4 = x^2 * x^2 ; x^6 = x^2 * x^4 ; x^7 = x * x^6

%Y See A003313 for the minimal number of multiplications to evaluate x^n.

%Y See A001190 for the total number of evaluation schemes for x^n (regardless of the number of effective multiplications).

%Y A079300 gives the number of minimal chains (= sequences of powers of x) ending at x^n. This is actually a bit less than the number of evaluation schemes since two schemes may produce the same chain, like the first and second schemes in the example above, where the corresponding chain is (x^2, x^3, x^4, x^7).

%K nonn

%O 1,5

%A Laurent Thevenoux and _Christophe Mouilleron_, Feb 23 2011