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A247749 Number T(n,k) of lattice paths from (0,0) to (n,0) that do not go below the x-axis or above the diagonal x=y, consist of steps u=(1,1), U=(1,3), H=(1,0), d=(1,-1) and D=(1,-3) for which the area below the path is k; triangle T(n,k), n>=0, read by rows.

%I A247749 #14 Apr 29 2022 05:46:40
%S A247749 1,1,1,1,1,2,1,1,3,3,2,2,0,1,1,4,6,6,6,3,4,2,1,1,1,5,10,13,15,12,14,
%T A247749 15,9,12,5,5,1,1,1,6,15,24,32,33,37,46,40,43,34,28,23,16,10,5,2,1,1,7,
%U A247749 21,40,61,75,88,114,122,134,137,118,127,101,99,69,68,41,38,19,17,5,5,0,1
%N A247749 Number T(n,k) of lattice paths from (0,0) to (n,0) that do not go below the x-axis or above the diagonal x=y, consist of steps u=(1,1), U=(1,3), H=(1,0), d=(1,-1) and D=(1,-3) for which the area below the path is k; triangle T(n,k), n>=0, read by rows.
%H A247749 Alois P. Heinz, <a href="/A247749/b247749.txt">Rows n = 0..50, flattened</a>
%F A247749 Sum_{k>=0}     T(n,k) = A240904(n).
%F A247749 Sum_{k>=1} k * T(n,k) = A247748(n).
%e A247749 Triangle T(n,k) begins:
%e A247749 1;
%e A247749 1;
%e A247749 1, 1;
%e A247749 1, 2,  1;
%e A247749 1, 3,  3,  2,  2,  0,  1;
%e A247749 1, 4,  6,  6,  6,  3,  4,  2,  1,  1;
%e A247749 1, 5, 10, 13, 15, 12, 14, 15,  9, 12,  5,  5,  1,  1;
%e A247749 1, 6, 15, 24, 32, 33, 37, 46, 40, 43, 34, 28, 23, 16, 10, 5, 2, 1;
%p A247749 b:= proc(x, y) option remember; `if`(y<0 or x<y, 0, `if`(x=0, 1,
%p A247749       expand(add(z^(y+j/2)*b(x-1, y+j), j=[-1, -3, 0, 1, 3]))))
%p A247749     end:
%p A247749 T:= n-> (p->seq(coeff(p, z, i), i=0..degree(p)))(b(n, 0)):
%p A247749 seq(T(n), n=0..10);
%t A247749 b[x_, y_] := b[x, y] = If[y < 0 || x < y, 0, If[x == 0, 1,
%t A247749      Expand[Sum[z^(y+j/2)*b[x-1, y+j], {j, {-1, -3, 0, 1, 3}}]]]];
%t A247749 T[n_] := CoefficientList[b[n, 0], z];
%t A247749 Table[T[n], {n, 0, 10}] // Flatten (* _Jean-François Alcover_, Apr 29 2022, after _Alois P. Heinz_ *)
%Y A247749 Cf. A240904, A247748.
%K A247749 nonn,tabf
%O A247749 0,6
%A A247749 _Alois P. Heinz_, Sep 23 2014