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A316773
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Triangle read by rows: T(n,m) = Sum_{k=m+1..n} (n-1)!/(k-1)!*binomial(2*n-k-1, n-1)*E(k,m) where E(n,m) is Euler's triangle A173018, T(0,0) = 1, n >= m >= 0.
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1
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1, 1, 0, 3, 1, 0, 19, 10, 1, 0, 193, 119, 23, 1, 0, 2721, 1806, 466, 46, 1, 0, 49171, 34017, 10262, 1502, 87, 1, 0, 1084483, 770274, 255795, 47020, 4425, 162, 1, 0, 28245729, 20429551, 7235853, 1539939, 193699, 12525, 303, 1, 0, 848456353, 621858526, 230629024, 54314242, 8273758, 755170, 34912, 574, 1, 0
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OFFSET
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0,4
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COMMENTS
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T(n,m) is the number of labeled binary trees of size n with m ascents on the left branch.
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LINKS
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FORMULA
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E.g.f.: Sum_{n >= m >= 0} T(n, m)/n! * x^n * y^m = E(C(x),y) = (y-1)/(y-exp(C(x)*(y-1))), where E(x,y) is an e.g.f. for Euler's triangle A173018.
T(n,m) = Sum_{k = m+1..n} C(n,k)*E(k,m)*P(n,n-k), T(0,0)=1, where C(n,m) is the transposed Catalan's triangle A033184, E(n,m) is Euler's triangle A173018, and P(n,m) is the number of k-permutations of n A008279.
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EXAMPLE
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Triangle begins:
--------------------------------------------------------------------------
n\k| 0 1 2 3 4 5 6 7 8 9
------+-------------------------------------------------------------------
0 | 1
1 | 1 0
2 | 3 1 0
3 | 19 10 1 0
4 | 193 119 23 1 0
5 | 2721 1806 466 46 1 0
6 | 49171 34017 10262 1502 87 1 0
7 | 1084483 770274 255795 47020 4425 162 1 0
8 | 28245729 20429551 7235853 1539939 193699 12525 303 1 0
9 | 848456353 621858526 230629024 54314242 8273758 755170 34912 574 1 0
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MAPLE
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T := (n, m) -> `if`(n=0, 1, add((n-1)!/(k-1)!*binomial(2*n-k-1, n-1)*
combinat:-eulerian1(k, m), k = m+1..n)):
for n from 0 to 6 do seq(T(n, k), k=0..n) od; # Peter Luschny, Sep 04 2020
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MATHEMATICA
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Table[Boole[n == 0] + Sum[(n - 1)!/(k - 1)!*Binomial[2 n - k - 1, n - 1]*Sum[(-1)^j*(m + 1 - j)^k*Binomial[k + 1, j], {j, 0, m}], {k, m + 1, n}], {n, 0, 8}, {m, 0, n}] // Flatten (* Michael De Vlieger, Sep 04 2020 *)
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PROG
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(Maxima)
T(n, m):=if m>n then 0 else if n=0 then 1 else sum((n-1)!/(k-1)!*binomial(2*n-k-1, n-1)*sum((-1)^j*(m+1-j)^k*binomial(k+1, j), j, 0, m), k, m+1, n);
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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