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%I #8 Feb 19 2015 05:45:03
%S 1,0,1,0,1,1,1,0,2,1,1,2,1,3,1,2,4,3,3,4,1,5,6,9,5,6,5,1,10,15,15,16,
%T 9,10,6,1,22,33,33,32,26,16,15,7,1,50,71,78,66,60,41,27,21,8,1,113,
%U 163,171,158,125,103,64,43,28,9,1,260,374,391,360,295,225,167,99,65,36,10,1
%N Triangle read by rows: T(n,k) is the number of weighted lattice paths B(n) having a total of k h- and H-steps at level 0.
%C B(n) is the set of lattice paths of weight n that start in (0,0), end on the horizontal axis and never go below this axis, whose steps are of the following four kinds: h = (1,0) of weight 1, H = (1,0) of weight 2, u = (1,1) of weight 2, and d = (1,-1) of weight 1. The weight of a path is the sum of the weights of its steps.
%C Row n contains n+1 entries.
%C Sum of entries in row n is A004148(n+1) (the 2ndary structure numbers).
%C T(n,0) = A166300(n).
%C Sum(k*T(n,k), k=0..n) = A247300(n)
%H Alois P. Heinz, <a href="/A247299/b247299.txt">Rows n = 0..140, flattened</a>
%H 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 G.f. G = 1/(1 - t*z - t*z^2 - z^3*g), where g is given by g = 1 + z*g + z^2*g + z^3*g^2.
%e Row 3 is 1,0,2,1 because B(3) = {ud, hH, Hh, hhh}.
%e Triangle starts:
%e 1;
%e 0,1;
%e 0,1,1;
%e 1,0,2,1;
%e 1,2,1,3,1;
%e 2,4,3,3,4,1;
%p eqg := g = 1+z*g+z^2*g+z^3*g^2: g := RootOf(eqg, g): G := 1/(1-t*z-t*z^2-z^3*g): Gser := simplify(series(G, z = 0, 16)): for n from 0 to 13 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 13 do seq(coeff(P[n], t, k), k = 0 .. n) end do; # yields sequence in triangular form
%p # second Maple program:
%p b:= proc(n, y) option remember; `if`(y<0 or y>n or n<0, 0,
%p `if`(n=0, 1, expand(`if`(y=0, x, 1)*(b(n-1, y)+
%p b(n-2, y)) +b(n-2, y+1) +b(n-1, y-1))))
%p end:
%p T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n, 0)):
%p seq(T(n), n=0..14); # _Alois P. Heinz_, Sep 17 2014
%t b[n_, y_] := b[n, y] = If[y<0 || y>n || n<0, 0, If[n == 0, 1, Expand[If[y == 0, x, 1]*(b[n-1, y] + b[n-2, y]) + b[n-2, y+1] + b[n-1, y-1]]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 0, n}]][b[n, 0]]; Table[T[n], {n, 0, 14}] // Flatten (* _Jean-François Alcover_, Feb 19 2015, after _Alois P. Heinz_ *)
%Y Cf. A004148, A166300, A247300,
%K nonn,tabl
%O 0,9
%A _Emeric Deutsch_, Sep 17 2014
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