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A300953 Number T(n,k) of Dyck paths of semilength n such that 2*k is the difference between the area above the path and the area below the path, measured within the smallest enclosing rectangle based on the x-axis; triangle T(n,k), n>=0, -floor((n-1)^2/4) <= k <= floor((n-1)^2/4), read by rows.

%I A300953 #28 May 09 2018 09:32:17
%S A300953 1,1,2,1,2,2,2,0,7,0,5,1,2,3,6,7,8,6,6,3,2,0,9,0,20,0,35,0,34,0,25,0,
%T A300953 7,1,2,4,8,10,17,23,30,38,43,46,48,42,41,26,26,12,8,4,2,0,11,0,29,0,
%U A300953 63,0,115,0,176,0,238,0,255,0,230,0,169,0,92,0,41,0,9
%N A300953 Number T(n,k) of Dyck paths of semilength n such that 2*k is the difference between the area above the path and the area below the path, measured within the smallest enclosing rectangle based on the x-axis; triangle T(n,k), n>=0, -floor((n-1)^2/4) <= k <= floor((n-1)^2/4), read by rows.
%H A300953 Alois P. Heinz, <a href="/A300953/b300953.txt">Rows n = 0..50, flattened</a>
%H A300953 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path#Counting_lattice_paths">Counting lattice paths</a>
%F A300953 Sum_{k = -floor((n-1)^2/4)..floor((n-1)^2/4)} k * T(n,k) = A300996(n).
%F A300953 T(n,-floor((n-1)^2/4)) = A040001(n).
%F A300953 T(n, floor((n-1)^2/4)) = A026741(n+1) for n > 2.
%F A300953 T(n,k) = 0 iff n is even and k is odd or abs(k) > floor(n*(n-1)/6).
%e A300953               .______.
%e A300953               | /\/\ |  ,  rectangle area: 12, above path area: 5,
%e A300953 T(3,-1) = 1:  |/____\|  ,  below path area: 7, difference: (5-7) = 2 * (-1).
%e A300953 .
%e A300953                  /\
%e A300953                 /  \
%e A300953 T(3,0) = 2:    /    \   /\/\/\  .
%e A300953 .
%e A300953                 /\         /\
%e A300953 T(3,1) = 2:    /  \/\   /\/  \  .
%e A300953 .
%e A300953 Triangle T(n,k) begins:
%e A300953 :                                   1                                    ;
%e A300953 :                                   1                                    ;
%e A300953 :                                   2                                    ;
%e A300953 :                               1,  2,  2                                ;
%e A300953 :                           2,  0,  7,  0,  5                            ;
%e A300953 :                   1,  2,  3,  6,  7,  8,  6,  6,  3                    ;
%e A300953 :           2,  0,  9,  0, 20,  0, 35,  0, 34,  0, 25,  0,  7            ;
%e A300953 :  1, 2, 4, 8, 10, 17, 23, 30, 38, 43, 46, 48, 42, 41, 26, 26, 12, 8, 4  ;
%Y A300953 Row sums give A000108.
%Y A300953 Column k=0 gives A300952.
%Y A300953 Cf. A002620, A026741, A040001, A129182, A239927, A300322, A300996.
%K A300953 nonn,tabf
%O A300953 0,3
%A A300953 _Alois P. Heinz_, Mar 16 2018