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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 (0<=k<=n).
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5
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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
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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COMMENTS
| 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).
Triangular array in A011117 transposed . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Mar 16 2005
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(a(n,k)*x^k,k=0..n) 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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FORMULA
| G.f.=2/[1+z+sqrt(1-6z+z^2)-2zt]
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 DELEHAM, 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. W. Lang, Sep 12 2005.
a(n, k)=((k+1)/(n-k))*sum(binomial(n-k, p)*binomial(n+p, p-1), p=1..(n-k)) 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. W. Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), 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). [From Paul Barry (pbarry(AT)wit.ie), 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),k,1,n-m+1), n>=m>1 [From Vladimir Kruchinin, Mar 17 2011]
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EXAMPLE
| Triangle starts:
1;
1,1;
3,2,1;
11,7,3,1;
45,28,12,4,1;
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).
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MAPLE
| 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
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PROG
| (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)} (from Paul D. Hanna (pauldhanna(AT)juno.com), Mar 30 2005)
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CROSSREFS
| Row sums are the large Schroeder numbers (A006318). Column 0 yields the little Schroeder numbers (A001003).
Cf. A104967 (matrix inverse).
Sequence in context: A116071 A077756 A115080 * A123513 A117442 A184182
Adjacent sequences: A104216 A104217 A104218 * A104220 A104221 A104222
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KEYWORD
| nonn,tabl
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 14 2005
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