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A104219
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Triangle read by rows: T(n,k) is number of Schroeder paths of length 2n and having k peaks at height 1, for 0 <= k <= n.
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9
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1, 1, 1, 3, 2, 1, 11, 7, 3, 1, 45, 28, 12, 4, 1, 197, 121, 52, 18, 5, 1, 903, 550, 237, 84, 25, 6, 1, 4279, 2591, 1119, 403, 125, 33, 7, 1, 20793, 12536, 5424, 1976, 630, 176, 42, 8, 1, 103049, 61921, 26832, 9860, 3206, 930, 238, 52, 9, 1, 518859, 310954, 134913, 49912
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OFFSET
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0,4
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COMMENTS
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A Schroeder path is a lattice path starting from (0,0), ending at a point on the x-axis, consisting only of steps U = (1,1), D = (1,-1) and H = (2,0) and never going below the x-axis. Schroeder paths are counted by the large Schroeder numbers (A006318).
This is an example of a Riordan (lower triangular) matrix. See the Shapiro et al. reference quoted under A053121. More precisely, this ordinary convolution triangle belongs to the Bell subgroup of the Riordan group. In the Shapiro et al. notation this is a Bell matrix (g(x), x*g(x)) with g(x) = (1+x-sqrt(1-6*x+x^2))/(4*x), the o.g.f. of A001003(n), n >= 0.
The g.f. for the row polynomials p(n,x) = Sum_{k=0..n} a(n,k)*x^k is g(y)/(1-x*y*g(y)) = (1-2*x*y+y-sqrt(1-6*y+y^2))/(2*y*(2-x-x*y+x^2*y)).
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LINKS
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FORMULA
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G.f.: 2/(1+z+sqrt(1-6*z+z^2)-2*z*t).
Another version of the triangle T(n, k), 0 <= k <= n, read by rows; given by [0, 1, 2, 1, 2, 1, 2, 1, 2, 1, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...] = 1; 0, 1; 0, 1, 1; 0, 3, 2, 1; 0, 11, 7, 3, 1; 0, 45, 28, 12, 4, 1; ... where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 16 2005
a(n, k) = (k+1)*hypergeom([1-n+k, n+2], [2], -1) if n > k; a(n, n)=1; a(n, k)=0 if n < k. - Wolfdieter Lang, Sep 12 2005
a(n, k) = ((k+1)/(n-k))*Sum_{p=1..n-k} binomial(n-k, p)*binomial(n+p, p-1) if n > k; a(n, n)=1; a(n, k)=0 if n < k. Proof with Lagrange's inversion theorem based on eq. y = 1+x*(1-2/y) where y=1/g(x), with g(x) the o.g.f. of A001003(n), n >= 0. Use G(k;y):=1/y^(k+1), k >= 0 to find the coefficients a(n, k) of x^n of G(k;1/g(x)). For this method see also the Larcombe and French paper on Catalan convolutions quoted under A033184. - Wolfdieter Lang, Sep 12 2005
G.f.: 1/(1-x*y-x/(1-x-x/(1-x-x/(1-x-x/(1-x-x/(1-... (continued fraction). - Paul Barry, Feb 01 2009
T((m+1)*n+r-1,m*n+r-1)*r/(m*n+r) = Sum_{k=1..n} (k/n)*T((m+1)*n-k-1,m*n-1)*(r+k,r), n >= m > 1, also T(n-1,m-1) = (m/n)*Sum_{k=1..n-m+1} k*A001003(k-1)*T(n-k-1,m-2), n >= m > 1. - Vladimir Kruchinin, Mar 17 2011
T(n, k) = (-1)^(n - k)*binomial(n, k)*hypergeom([k - n, n + 1], [k + 2], 2). - Peter Luschny, Jan 08 2018
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EXAMPLE
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Triangle starts:
[0] 1;
[1] 1, 1;
[2] 3, 2, 1;
[3] 11, 7, 3, 1;
[4] 45, 28, 12, 4, 1;
[5] 197, 121, 52, 18, 5, 1;
[6] 903, 550, 237, 84, 25, 6, 1;
T(3,1)=7 because we have HH(UD),H(UD)H,(UD)HH,UUDD(UD),(UD)UUDD,(UD)UHD, and
UHD(UD) (the peaks UD at height 1 are shown between parentheses).
Production matrix begins:
1, 1;
2, 1, 1;
4, 2, 1, 1;
8, 4, 2, 1, 1;
16, 8, 4, 2, 1, 1;
32, 16, 8, 4, 2, 1, 1;
64, 32, 16, 8, 4, 2, 1, 1; (End)
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MAPLE
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G:=2/(1+z+sqrt(1-6*z+z^2)-2*z*t): Gser:=simplify(series(G, z=0, 13)): P[0]:=1: for n from 1 to 13 do P[n]:=coeff(Gser, z^n) od: for n from 0 to 11 do seq(coeff(t*P[n], t^k), k=1..n+1) od; # yields sequence in triangular form
# Alternatively:
T_row := proc(n) local c, f, s;
c := N -> hypergeom([1-N, N+2], [2], -1);
f := n -> 1+add(simplify(c(i))*x^i, i=1..n):
s := j -> coeff(series(f(j)/(1-x*t*f(j)), x, j+1), x, j):
seq(coeff(s(n), t, j), j=0..n) end:
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MATHEMATICA
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T[n_, k_] := (-1)^(n - k) Binomial[n, k] Hypergeometric2F1[k - n, n + 1, k + 2, 2];
Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Peter Luschny, Jan 08 2018 *)
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PROG
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(PARI) {T(n, k)=local(X=x+x*O(x^n), Y=y+y*O(y^k)); polcoeff(polcoeff(2/(1+X+sqrt(1-6*X+X^2)-2*X*Y), n, x), k, y)} \\ Paul D. Hanna, Mar 30 2005
(Sage)
@cached_function
def prec(n, k):
if k==n: return 1
if k==0: return 0
return prec(n-1, k-1)+sum(prec(n, k+i-1) for i in (2..n-k+1))
return [prec(n, k) for k in (1..n)]
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CROSSREFS
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Row sums are the large Schroeder numbers (A006318). Column 0 yields the little Schroeder numbers (A001003).
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KEYWORD
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AUTHOR
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STATUS
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approved
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