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A091156
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Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n, having k long ascents (i.e., ascents of length at least 2). Rows are of length 1,1,2,2,3,3,... .
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5
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1, 1, 1, 1, 1, 4, 1, 11, 2, 1, 26, 15, 1, 57, 69, 5, 1, 120, 252, 56, 1, 247, 804, 364, 14, 1, 502, 2349, 1800, 210, 1, 1013, 6455, 7515, 1770, 42, 1, 2036, 16962, 27940, 11055, 792, 1, 4083, 43086, 95458, 57035, 8217, 132, 1, 8178, 106587, 305812, 257257
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OFFSET
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0,6
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COMMENTS
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Also number of ordered trees with n edges, having k branch nodes (i.e., vertices of outdegree at least 2).
Also number of Łukasiewicz paths of length n having k fall steps (1,-1) that start at an odd level. A Łukasiewicz path of length n is a path in the first quadrant from (0,0) to (n,0) using rise steps (1,k) for any positive integer k, level steps (1,0) and fall steps (1,-1) (see R. P. Stanley, Enumerative Combinatorics, Vol. 2, Cambridge Univ. Press, Cambridge, 1999, p. 223, Exercise 6.19w; the integers are the slopes of the steps). Example: T(4,2)=2 because we have U(D)U(D) and U(3)(D)D(D), where U=(1,1), D=(1,-1), U(3)=(1,3) and the fall steps that start at an odd level are shown between parentheses. Row n has 1+floor(n/2) terms. Row sums are the Catalan numbers (A000108). T(2n,n)=A000108(n). T(2n+1,n)=A001791(n+1)=binomial(2n+2,n). - Emeric Deutsch, Jan 06 2005
Also number of Dyck paths of semilength n with k UUD's - I. Tasoulas (jtas(AT)unipi.gr), Feb 19 2006
T(n,k) = number of Dyck n-paths whose decomposition into 2-step subpaths contains k UUs. For example, T(4,2)=2 counts UU|UU|DD|DD, UU|DD|UU|DD (vertical bars indicate path decomposition). - David Callan, Jun 07 2006
T(n,k) = number of binary trees on n-1 edges containing k right edges whose child vertex has no right child. Under Knuth's "natural" correspondence, such a vertex in binary (n-1)-tree ~ a vertex of outdegree >=2 in ordered n-tree. - David Callan, Sep 25 2006
T(n,k) = number of binary trees on n-1 edges containing k left edges whose child vertex has no left child. Under "natural" correspondence, such a vertex in binary (n-1)-tree ~ a leaf edge with no left neighbor edge and not incident to the root in ordered n-tree ~ a UUD in Dyck n-path. - David Callan, Sep 25 2006
T(n,k) = number of permutations of length n avoiding 321 (classically) with k descents. - Andrew Baxter, May 17 2011.
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REFERENCES
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R. P. Stanley, Enumerative Combinatorics, Vol. 1, 1986; See Exercise 3.71(f).
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LINKS
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FORMULA
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T(n,k) = (1/(n+1)) * binomial(n+1, k) * Sum_{j=0..n-2k} binomial(k+j-1, k-1)*binomial(n+1-k, n-2k-j).
G.f. G(t, z) satisfies z*(1-z+t*z)*G^2 - G + 1 = 0.
T(n,k) = n!*(1+k)/((n-2*k)!*(1+k)!^2)*hypergeom([k,2*k-n], [k+2], -1). - Peter Luschny, Oct 16 2015
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EXAMPLE
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T(4,1) = 11 because among the 14 Dyck paths of semilength 4, the paths that do not have exactly one long ascent are UDUDUDUD (no long ascent), UUDDUUDD (two long ascents) and UUDUUDDD (two long ascents). Here U=(1,1) and D=(1,-1).
Triangle begins:
1;
1;
1, 1;
1, 4;
1, 11, 2;
1, 26, 15;
1, 57, 69, 5;
1, 120, 252, 56;
1, 247, 804, 364, 14;
1, 502, 2349, 1800, 210;
1, 1013, 6455, 7515, 1770, 42;
...
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MAPLE
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a := (n, k)->binomial(n+1, k)* add(binomial(k+j-1, k-1)*binomial(n+1-k, n-2*k-j), j=0..n-2*k)/(n+1); seq(seq(a(n, k), k=0..floor(n/2)), n=0..15);
seq(seq(simplify(n!*(1+k)/((n-2*k)!*(1+k)!^2)*hypergeom([k, 2*k-n], [k+2], -1)), k=0.. floor(n/2)), n=0..15); # Peter Luschny, Oct 16 2015
# alternative Maple program:
b:= proc(x, y) option remember; `if`(y>x or y<0, 0,
`if`(x=0, 1, expand(b(x-1, y)*`if`(y=0, 1, 2)*z+
b(x-1, y+1) +b(x-1, y-1))))
end:
T:= n-> (p-> seq(coeff(p, z, n-2*i), i=0..n/2))(b(n, 0)):
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MATHEMATICA
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T[n_, k_] := Binomial[n+1, k]*Sum[Binomial[k+j-1, k-1]*Binomial[n+1-k, n- 2*k-j], {j, 0, n-2*k}]/(n+1); Table[T[n, k], {n, 0, 15}, {k, 0, Floor[n/2 ]}] // Flatten (* Jean-François Alcover, Jan 31 2016 *)
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PROG
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(PARI)
tabf(nn) = {for(n=-1, nn, for(k=0, floor(n/2), if(binomial(n+1, k) * sum(j=0, n-2*k, binomial(k+j-1, k-1) * binomial(n+1-k, n-2*k-j))/(n+1)==0, print1("1, "), print1(binomial(n+1, k) * sum(j=0, n-2*k, binomial(k+j-1, k-1) * binomial(n+1-k, n-2*k-j))/(n+1), ", ")); ); print(); ); };
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CROSSREFS
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T(n,k) are rational multiples of A055151.
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KEYWORD
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nonn,tabf
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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