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A201080 Irregular triangle read by rows: number of shifted Schroeder paths of length n and area k.

%I A201080 #21 Jun 11 2018 11:52:46
%S A201080 1,1,1,1,1,1,2,1,1,1,1,3,3,4,3,3,3,1,1,1,1,3,3,5,6,8,9,11,12,11,10,7,
%T A201080 6,4,1,1,1,1,3,3,6,6,9,12,16,18,22,27,29,33,38,40,39,39,34,28,21,14,
%U A201080 10,5,1,1,1,1,3,3,6,6,10,13,18,22,28,35,41,50,61
%N A201080 Irregular triangle read by rows: number of shifted Schroeder paths of length n and area k.
%H A201080 Alois P. Heinz, <a href="/A201080/b201080.txt">Rows n = 0..40, flattened</a>
%H A201080 Brian Drake, <a href="http://dx.doi.org/10.1016/j.disc.2008.11.020">Limits of areas under lattice paths</a>, Discrete Math. 309 (2009), no. 12, 3936-3953. See Example 3.
%e A201080 Triangle begins
%e A201080 1
%e A201080 1 1
%e A201080 1 1 1 2 1
%e A201080 1 1 1 3 3 4 3 3 3 1
%e A201080 1 1 1 3 3 5 6 8 9 11 12 11 10 7 6 4 1
%e A201080 ...
%p A201080 b:= proc(x, y) option remember; expand(`if`(y>x or y<0, 0,
%p A201080       `if`(x=0, 1, b(x-1, y)*z^(2*y)+b(x, y-1)+`if`(y>0, add(
%p A201080        b(x-(2*j-1), y-1)*z^((2*y-1)*(2*j-1)), j=1..1+(x-y)/2), 0))))
%p A201080     end:
%p A201080 T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(n$2)):
%p A201080 seq(T(n), n=0..8);  # _Alois P. Heinz_, Feb 02 2018
%t A201080 b[x_, y_] := b[x, y] = Expand[If[y > x || y < 0, 0, If[x == 0, 1, b[x - 1, y]*z^(2*y) + b[x, y - 1] + If[y > 0, Sum[b[x - (2*j - 1), y - 1]*z^((2*y - 1)*(2*j - 1)), {j, 1, 1 + (x - y)/2}], 0]]]];
%t A201080 T[n_] := Function[p, Table[Coefficient[p, z, i], {i, 0, Exponent[p, z]}]][ b[n, n]];
%t A201080 Table[T[n], {n, 0, 8}] // Flatten (* _Jean-François Alcover_, Jun 11 2018, after _Alois P. Heinz_ *)
%Y A201080 Row sums give A133656.
%K A201080 nonn,tabf
%O A201080 0,7
%A A201080 _N. J. A. Sloane_, Nov 26 2011
%E A201080 More term from _Alois P. Heinz_, Feb 02 2018