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A129174 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n such that the sum of the peak-abscissae is k (0 <= k <= n^2). 3
1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 1, 2, 1, 2, 1, 2, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 2, 2, 3, 2, 4, 3, 4, 3, 4, 2, 3, 2, 2, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 2, 2, 4, 3, 5, 5, 7, 6, 9, 7, 9, 8, 9, 7, 9, 6, 7, 5, 5, 3, 4, 2, 2, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,27
COMMENTS
Row n contains 1+n^2 entries. Row sums are the Catalan numbers (A000108). Column sums yield A129528. T(n,n+k) = T(n,n^2-k) (i.e., rows are palindromic). Alternating row sums are (-1)^n*binomial(n,floor(n/2)) = A126930(n). Sum_{k=0..n^2} k*T(n,k) = n*binomial(2n-1,n-1) = A002457(n-1). T(n,k) = A129175(n,n-k) (i.e., except for the initial 0's, rows of A129174 and A129175 are the same).
REFERENCES
G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976.
LINKS
FindStat - Combinatorial Statistic Finder, The major index of a Dyck path.
FORMULA
The generating polynomial for row n is P[n](t) = t^n*binomial[2n,n]/[n+1], where [n+1]=1+t+t^2+...+t^n and binomial[2n,n] is a Gaussian polynomial (in t).
EXAMPLE
T(5,11)=3 because we have (i) UDUDUUUDDD with peak-abscissae 1,3,7, (ii) UUUDDUUDD with peak-abscissae 3,8 and (iii) UUUUDDUDDD with peak-abscissae 4,7; here U=(1,1) and D=(1,-1).
Triangle starts:
1;
0,1;
0,0,1,0,1;
0,0,0,1,0,1,1,1,0,1;
0,0,0,0,1,0,1,1,2,1,2,1,2,1,1,0,1;
...
MAPLE
br:=n->sum(q^i, i=0..n-1): f:=n->product(br(j), j=1..n): cbr:=(n, k)->f(n)/f(k)/f(n-k): P:=n->sort(expand(simplify(q^n*cbr(2*n, n)/br(n+1)))): for n from 0 to 7 do seq(coeff(P(n), q, k), k=0..n^2) od; # yields sequence in triangular form
# second Maple program:
b:= proc(x, y, t) option remember; `if`(y<0 or y>x, 0, `if`(x=0, 1,
expand(b(x-1, y+1, 1) +`if`(t=1, z^x, 1)*b(x-1, y-1, 0))))
end:
T:= n-> (p-> seq(coeff(p, z, i), i=0..n^2))(b(2*n, 0$2)):
seq(T(n), n=0..8); # Alois P. Heinz, Jun 10 2014
MATHEMATICA
b[x_, y_, t_] := b[x, y, t] = If[y<0 || y>x, 0, If[x == 0, 1, Expand[b[x-1, y+1, 1] + If[t == 1, z^x, 1]*b[x-1, y-1, 0]]]]; T[n_] := Function[{p}, Table[ Coefficient[p, z, i], {i, 0, n^2}]][b[2*n, 0, 0]]; Table[T[n], {n, 0, 8}] // Flatten (* Jean-François Alcover, May 26 2015, after Alois P. Heinz *)
CROSSREFS
Sequence in context: A212212 A212213 A214339 * A129175 A334377 A063053
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, Apr 20 2007
STATUS
approved

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Last modified February 28 09:01 EST 2024. Contains 370394 sequences. (Running on oeis4.)