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A246177 Triangle read by rows: T(n,k) is the number of weighted lattice paths in B(n) such that the area between the x-axis and the path is k.

%I A246177 #21 Mar 26 2018 12:40:21
%S A246177 1,1,2,3,1,5,2,1,8,5,3,1,13,10,8,4,2,21,20,18,12,7,3,1,34,38,39,30,22,
%T A246177 12,7,2,1,55,71,80,70,57,39,26,14,7,3,1,89,130,160,154,138,106,81,52,
%U A246177 34,18,10,4,2,144,235,312,327,315,267,220,163,118,78,49,28,16,7,3,1
%N A246177 Triangle read by rows: T(n,k) is the number of weighted lattice paths in B(n) such that the area between the x-axis and the path is k.
%C A246177 The members of B(n) are paths of weight n that start at (0,0), end on but never go below the horizontal axis, and whose steps are of the following four kinds: an (1,0)-step with weight 1, an (1,0)-step with weight 2, a (1,1)-step with weight 2, and a (1,-1)-step with weight 1. The weight of a path is the sum of the weights of its steps.
%C A246177 Apparently, number of terms in row n is 1+floor(n^2/8).
%C A246177 Sum of entries in row n is A004148(n+1) (the 2ndary structure numbers).
%C A246177 T(n,0) = A000045(n+1) (the Fibonacci numbers).
%C A246177 T(n,1) = A001629(n-1) (n>=1).
%H A246177 Alois P. Heinz, <a href="/A246177/b246177.txt">Rows n = 0..65, flattened</a>
%H A246177 M. Bona and A. Knopfmacher, <a href="http://dx.doi.org/10.1007/s00026-010-0060-7">On the probability that certain compositions have the same number of parts</a>, Ann. Comb., 14 (2010), 291-306.
%F A246177 The trivariate g.f. G=G(t,s,z), where t marks area, s marks length (=number of steps), and z marks weight, satisfies G = 1+szG+sz^2G+ts^2z^3G(t,ts,z)G. This follows at once from the fact that every nonempty path is of the form hC or HC or UCDC, where h denotes a (1,0)-step of weight 1, H denotes a (1,0)-step of weight 2, U denotes a (1,1)-step, D denotes  a (1,-1)-step, and the C's denote paths, not necessarily the same. From the equation one can find G(t,s,z) as a continued fraction (the Maple program makes use of this).
%e A246177 Row 3 is 3,1; indeed, B(3) consists of the paths hhh, hH, Hh, UD with areas 0,0,0,1, respectively.
%e A246177 Triangle starts:
%e A246177    1;
%e A246177    1;
%e A246177    2;
%e A246177    3,   1;
%e A246177    5,   2,   1;
%e A246177    8,   5,   3,   1;
%e A246177   13,  10,   8,   4,   2;
%e A246177   21,  20,  18,  12,   7,   3,  1;
%e A246177   34,  38,  39,  30,  22,  12,  7,  2,  1;
%e A246177   55,  71,  80,  70,  57,  39, 26, 14,  7,  3,  1;
%e A246177   89, 130, 160, 154, 138, 106, 81, 52, 34, 18, 10, 4, 2;
%p A246177 g := 1/(1-z-z^2-t*z^3*A[1]): for j to 15 do A[j] := 1/(1-t^j*z-t^j*z^2-t^(2*j+1)*z^3*A[j+1]) end do: gser := simplify(series(g, z = 0, 20)): for n from 0 to 17 do P[n] := sort(coeff(gser, z, n)) end do: for n from 0 to 17 do seq(coeff(P[n], t, j), j = 0 .. floor((1/8)*n^2)) end do; # yields sequence in triangular form
%p A246177 # second Maple program:
%p A246177 b:= proc(n, y) option remember; `if`(y<0 or y>n, 0, `if`(n=0, 1,
%p A246177       expand(b(n-1, y)*x^y +`if`(n>1, b(n-2, y)*x^y+b(n-2, y+1)*
%p A246177       x^(y+1/2), 0) +b(n-1, y-1)*x^(y-1/2))))
%p A246177     end:
%p A246177 T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0)):
%p A246177 seq(T(n), n=0..20);  # _Alois P. Heinz_, Aug 20 2014
%t A246177 b[n_, y_] := b[n, y] = If[y<0 || y>n, 0, If[n == 0, 1, Expand[b[n-1, y]*x^y + If[n>1, b[n-2, y]*x^y + b[n-2, y+1]*x^(y+1/2), 0] + b[n-1, y-1]*x^(y-1/2)]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, 0]]; Table[T[n], {n, 0, 20}] // Flatten (* _Jean-François Alcover_, May 27 2015, after _Alois P. Heinz_ *)
%Y A246177 Cf. A004148, A000045, A001629.
%K A246177 nonn,tabf
%O A246177 0,3
%A A246177 _Emeric Deutsch_, Aug 20 2014