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A247294 Triangle read by rows: T(n,k) is the number of weighted lattice paths B(n) having a total of k uhd and uHd strings. 5
1, 1, 2, 4, 7, 1, 14, 3, 30, 7, 64, 18, 141, 43, 1, 316, 102, 5, 713, 249, 16, 1626, 608, 49, 3740, 1489, 143, 1, 8659, 3669, 400, 7, 20176, 9058, 1109, 29, 47274, 22407, 3046, 105, 111302, 55560, 8282, 357, 1, 263201, 138004, 22385, 1149, 9 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
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.
Row n contains 1 + floor(n/4) entries.
Sum of entries in row n is A004148(n+1) (the 2ndary structure numbers).
T(n,0) = A247295(n).
Sum(k*T(n,k), k=0..n) = A247296(n).
LINKS
M. Bona and A. Knopfmacher, On the probability that certain compositions have the same number of parts, Ann. Comb., 14 (2010), 291-306.
FORMULA
G.f. G = G(t,z) satisfies G = 1 + z*G + z^2*G + z^3*G*(G - z - z^2 + t*z + t*z^2).
EXAMPLE
T(6,1)=7 because we have uhdhh, huhdh, hhuhd, Huhd, uhdH, uHdh, and huHd.
Triangle starts:
1;
1;
2;
4;
7,1;
14,3;
30,7;
MAPLE
eq := G = 1+z*G+z^2*G+z^3*(G-z-z^2+t*z+t*z^2)*G: G := RootOf(eq, G): Gser := simplify(series(G, z = 0, 25)): for n from 0 to 22 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 22 do seq(coeff(P[n], t, k), k = 0 .. floor((1/4)*n)) end do; # yields sequence in triangular form
# second Maple program:
b:= proc(n, y, t) option remember; `if`(y<0 or y>n, 0, `if`(n=0, 1,
expand(b(n-1, y-1, 0)*`if`(t=2, x, 1)+b(n-1, y, `if`(t=1, 2, 0))
+`if`(n>1, b(n-2, y, `if`(t=1, 2, 0))+b(n-2, y+1, 1), 0))))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0$2)):
seq(T(n), n=0..20); # Alois P. Heinz, Sep 16 2014
MATHEMATICA
b[n_, y_, t_] := b[n, y, t] = If[y<0 || y>n, 0, If[n == 0, 1, Expand[b[n-1, y-1, 0]*If[t == 2, x, 1] + b[n-1, y, If[t == 1, 2, 0]] + If[n>1, b[n-2, y, If[t == 1, 2, 0]] + b[n-2, y+1, 1], 0]]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, 0, 0]]; Table[T[n], {n, 0, 20}] // Flatten (* Jean-François Alcover, May 27 2015, after Alois P. Heinz *)
CROSSREFS
Sequence in context: A110317 A098073 A118390 * A202848 A202841 A247290
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, Sep 16 2014
STATUS
approved

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Last modified February 29 13:45 EST 2024. Contains 370425 sequences. (Running on oeis4.)