%I #17 Nov 03 2023 17:59:36

%S 1,0,1,1,2,1,0,2,4,1,1,5,8,5,1,0,3,14,15,6,1,1,9,25,32,21,7,1,0,4,32,

%T 62,56,28,8,1,1,14,56,109,122,84,36,9,1,0,5,60,170,242,210,120,45,10,

%U 1,1,20,105,275,436,457,330,165,55,11,1,0,6,100,375,732,912,792,495,220,66,12,1

%N Triangle T(r,c) of winning binary "same game" templates.

%C T(r,c) is the number of winning templates with length r and minimum matching string length c; equivalently, ternary digits totaling r+c. For a definition and row sums 1,1,4,7,20, etc. see A066345. For antidiagonal sums 1,0,2,2,4,9, etc. see A066346.

%H Sean A. Irvine, <a href="https://github.com/archmageirvine/joeis/blob/master/src/irvine/oeis/a066/A066709.java">Java program</a> (github)

%F A035615(n) = 2 * Sum_{r=1..n-1, c=1..min(r,n-r)} T(r,c) * P(n-r,c) where P(n-r,c) = C(n-r-1,c-1) = (n-r-1)!/((n-r-c-2)!*(c-1)!).

%e Rows:

%e 1;

%e 0,1;

%e 1,2,1;

%e 0,2,4,1;

%e 1,5,8,5,1;

%e 0,3,14,15,6,1; ...

%e a(17) = T(6,2) = 3 winning templates with length 6 and total 8 = 6+2: 211211, 121121, 112112.

%e A035615(6) = 2*( 1*1+0*1+1*3+1*1+2*2+1*1+1*1+0*1+2*1+1*1 ) = 2*13 = 26.

%Y Cf. A035615, A066345, A066346, A007318.

%K nonn,tabl

%O 1,5

%A _Frank Ellermann_, Dec 31 2001

%E More terms from _Sean A. Irvine_, Nov 03 2023