|
%I #35 Sep 27 2023 17:34:55
%S 1,1,1,1,2,1,1,1,1,3,3,3,4,3,2,1,1,1,1,4,6,7,10,11,10,9,8,7,5,4,3,2,1,
%T 1,1,1,5,10,14,21,28,31,33,34,34,31,27,25,22,17,14,13,10,7,5,4,3,2,1,
%U 1,1,1,6,15,25,40,60,77,92,106,117,122,121,120,116,107,98,91,82,71,62,54,45
%N Triangle read by rows: T(n, k) is the number of Schroeder paths of semilength n such that the area between the x-axis and the path is k (n >= 0; 0 <= k <= n^2).
%C A Schroeder path of semilength n is a lattice path from (0,0) to (2n,0) consisting of U = (1,1), D = (1,-1) and H = (2,0) steps and never going below the x-axis.
%C Row n has 1+n^2 terms.
%C Row sums are the large Schroeder numbers (A006318).
%H Alois P. Heinz, <a href="/A129179/b129179.txt">Rows n = 0..32, flattened</a>
%F G.f.: G(t,z) satisfies G(t,z) = 1 + z*G(t,z) + t*z*G(t,t^2*z)*G(t,z).
%F Sum_{k>=0} k*T(n,k) = A129180(n).
%F From _Peter Bala_, Aug 02 2019: (Start)
%F O.g.f. as a continued fraction: (t marks the area and z marks the semilength of the path)
%F G(t,z) = 1/(1 - z - t*z/(1 - t^2*z - t^3*z/(1 - t^4*z - t^5*z/(1 - t^6*z - (...) )))) = 1 + z*(1 + t) + z^2*(1 + 2*t + t^2 + t^3 + t^4) + ....
%F G(t,z) = 1/(1 - (1 + t)*z/(1 - t^3*z/(1 - (t^2 + t^5)*z/(1 - t^7*z/(1 - (t^4 + t^9)*z/(1 - t^11*z/( (...) ))))))).
%F O.g.f. as a ratio of q-series: N(t,z)/D(t,z), where N(t,z) = Sum_{n >= 0} (-1)^n*t^(2*n^2+n)*z^n/( (Product_{k = 1..n} 1 - t^(2*k)) * (Product_{k = 1..n+1} 1 - t^(2*k-2)*z) ) and D(t,z) = Sum_{n >= 0} (-1)^n*t^(2*n^2-n)*z^n/( (Product_{k = 1..n} 1 - t^(2*k)) * (Product_{k = 1..n} 1 - t^(2*k-2)*z) ). (End)
%F T(n, k) = [z^k] R(n, 0) for n >= 0, k >= 0 where R(n, q) = Sum_{j=0..q + (q mod 2) + 1} z^j*R(n-1, j) for n > 0, q >= 0 with R(0, q) = 1 for q >= 0. - _Mikhail Kurkov_, Aug 03 2023
%e T(3,5) = 3 because we have UDUUDD, UUDDUD and UHHD.
%e Triangle starts:
%e 1;
%e 1,1;
%e 1,2,1,1,1;
%e 1,3,3,3,4,3,2,1,1,1;
%e 1,4,6,7,10,11,10,9,8,7,5,4,3,2,1,1,1;
%p G:=1/(1-z-t*z*g[1]): for i from 1 to 11 do g[i]:=1/(1-t^(2*i)*z-t^(2*i+1)*z*g[i+1]) od: g[12]:=0: Gser:=simplify(series(G,z=0,13)): for n from 0 to 11 do P[n]:=sort(coeff(Gser,z,n)) od: for n from 0 to 6 do seq(coeff(P[n],t,j),j=0..n^2) od; # yields sequence in triangular form
%p # second Maple program:
%p b:= proc(x, y) option remember; `if`(y>x or y<0, 0,
%p `if`(x=0, 1, expand(b(x-1, y-1)*z^(y-1/2)
%p +b(x-2, y)*z^(2*y) +b(x-1, y+1)*z^(y+1/2))))
%p end:
%p T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(2*n, 0)):
%p seq(T(n), n=0..7); # _Alois P. Heinz_, May 27 2015
%p R := proc(n, q) option remember; local j; if n <= 0 then 1 else
%p add(z^j*R(n-1, j), j = 0..q + irem(q, 2) + 1) fi end:
%p Trow := n -> local k; seq(coeff(R(n, 0), z, k), k = 0..n^2):
%p seq(Trow(n), n = 0..7); # after _Mikhail Kurkov_, _Peter Luschny_, Sep 27 2023
%t b[x_, y_] := b[x, y] = If[y>x || y<0, 0, If[x==0, 1, Expand[b[x-1, y-1]*z^(y-1/2) + b[x-2, y]*z^(2*y) + b[x-1, y+1]*z^(y+1/2)]]]; T[n_] := Function[{p}, Table[ Coefficient[p, z, i], {i, 0, Exponent[p, z]}]][b[2*n, 0]]; Table[T[n], {n, 0, 7}] // Flatten (* _Jean-François Alcover_, Jun 29 2015, after _Alois P. Heinz_ *)
%o (PARI) rows_upto(n)=my(v1, v2, v3, z='z); v1=vector(2*n+1, i, 1); v2=v1; v3=vector(n+1, i, 0); v3[1]=[1]; for(i=1, n, for(q=0, 2*(n-i), v2[q+1]=sum(j=0, q + q%2 + 1, z^j*v1[j+1])); v1=v2; v3[i+1]=Vecrev(v1[1])); v3 \\ _Mikhail Kurkov_, Aug 03 2023
%Y Cf. A006318 (row sums), A129180, A326676.
%K nonn,tabf
%O 0,5
%A _Emeric Deutsch_, Apr 08 2007
|
