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A209919 Triangle read by rows: T(n,k), 0 <= k <= n-1, = number of 2-divided binary sequences of length n which are 2-divisible in exactly k ways.

%I A209919 #44 May 06 2021 08:11:01
%S A209919 0,3,1,4,2,2,6,3,4,3,8,6,6,6,6,14,9,11,10,11,9,20,18,18,18,18,18,18,
%T A209919 36,30,33,30,34,30,33,30,60,56,56,58,56,56,58,56,56,108,99,105,99,105,
%U A209919 100,105,99,105,99,188,186,186,186,186,186,186,186,186,186,186,352,335,344,338,346,335,348,335,346,338,344,335,632,630,630,630,630,630,630,630,630,630,630,630,630,1182,1161,1179,1161,1179,1161,1179,1162,1179,1161,1179,1161,1179,1161,2192,2182,2182,2188,2182,2184,2188,2182,2182,2188,2184,2182,2188,2182,2182
%N A209919 Triangle read by rows: T(n,k), 0 <= k <= n-1, = number of 2-divided binary sequences of length n which are 2-divisible in exactly k ways.
%C A209919 Computed by _David Scambler_.
%C A209919 See A210109 for further information.
%C A209919 Omitting the leading column, triangle has mirror symmetry.
%C A209919 Speculation: T(2n+1,2)=T(2n+1,1); T(2n,2)=T(2n,1)+T(n,1); T(3n+1,3)=T(3n+1,1); T(3n+2,3)=T(3n+2,1); T(3n,3)=T(3n,1)+T(n,1) and similar "lagged modulo sums" for T(4n+i,4)=T(4n+i,2), 0<i<=3; T(4n,4)=T(4n,2)+T(n,1); T(5n+i,5)=T(5n+i,1), 0<i<=4; T(5n,5)=T(5n,1)+T(n,1). - _R. J. Mathar_, Mar 27 2012
%C A209919 Right border appears to be A059966. - _Michel Marcus_, Apr 26 2013
%e A209919 Triangle begins:
%e A209919 n  k=0  k=1  k=2  k=3  k=4  k=5  k=6  k=7  k=8  k=9  k=10 k=11 k=12 k=13 k=14
%e A209919 1  1
%e A209919 2  3    1
%e A209919 3  4    2    2
%e A209919 4  6    3    4    3
%e A209919 5  8    6    6    6    6
%e A209919 6  14   9    11   10   11   9
%e A209919 7  20   18   18   18   18   18   18
%e A209919 8  36   30   33   30   34   30   33   30
%e A209919 9  60   56   56   58   56   56   58   56   56
%e A209919 10 108  99   105  99   105  100  105  99   105  99
%e A209919 11 188  186  186  186  186  186  186  186  186  186  186
%e A209919 12 352  335  344  338  346  335  348  335  346  338  344  335
%e A209919 13 632  630  630  630  630  630  630  630  630  630  630  630  630
%e A209919 14 1182 1161 1179 1161 1179 1161 1179 1162 1179 1161 1179 1161 1179 1161
%e A209919 15 2192 2182 2182 2188 2182 2184 2188 2182 2182 2188 2184 2182 2188 2182 2182...
%Y A209919 First column is A000031, second column is conjectured to be A001037. Row sums = 2^n.
%K A209919 nonn,tabl
%O A209919 1,2
%A A209919 _N. J. A. Sloane_, Mar 21 2012