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A101974
Triangle read by rows: number of Dyck paths of semilength n with k peaks before the first return (1<= k <n).
1
1, 2, 4, 1, 9, 4, 1, 23, 11, 7, 1, 65, 27, 28, 11, 1, 197, 66, 87, 62, 16, 1, 626, 170, 239, 250, 122, 22, 1, 2056, 471, 627, 829, 630, 219, 29, 1, 6918, 1398, 1656, 2448, 2553, 1419, 366, 37, 1, 23714, 4381, 4554, 6803, 8813, 6979, 2917, 578, 46, 1, 82500, 14282
OFFSET
1,2
REFERENCES
E. Deutsch, Dyck path enumeration, Discrete Math., 204, 1999, 167-202.
FORMULA
T(n, 1)=sum(c(i), i=0..n-1), T(n, k)=sum(c(j)*binomial(n-1-j, k-1)*binomial(n-1-j, k)/(n-1-j), j=0..n-2) for k>1, where c(i)=binomial(2i, i)/(i+1) (i=0, 1, ...) are the Catalan numbers (A000108);
G.f.=1+tzC(z)[1+r(t, z)], where C(z)=1+zC(z)^2 is the Catalan function and r(t, z)=z[1+r(t, z)][1+tr(t, z)] is the Narayana function.
EXAMPLE
T(4,2)=4 because we have U(UD)(UD)D|UD, U(UD)U(UD)DD|, UU(UD)D(UD)D| and
UU(UD)(UD)DD|, where U=(1,1), D=(1,-1) (the peaks before the first return | are shown between parentheses).
1
2
4 1
9 4 1
23 11 7 1
65 27 28 11 1
197 66 87 62 16 1
626 170 239 250 122 22 1
2056 471 627 829 630 219 29 1
6918 1398 1656 2448 2553 1419 366 37 1
23714 4381 4554 6803 8813 6979 2917 578 46 1
82500 14282 13231 18571 27362 28364 17206 5567 872 56 1
MAPLE
c:=n->binomial(2*n, n)/(n+1):
T:=proc(n, k) if k=1 then sum(c(i), i=0..n-1) else sum(c(j)*binomial(n-1-j, k-1)*binomial(n-1-j, k)/(n-1-j), j=0..n-2) fi end proc:
T(1, 1);
for n from 1 to 12 do seq(T(n, k), k=1..n-1) od; # yields the sequence in triangular form
CROSSREFS
Cf. A000108 (row sums), A014137 (column k=1), A014151 (column k=2), A101975.
Sequence in context: A321461 A092107 A114489 * A097607 A132893 A273896
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, Dec 22 2004
STATUS
approved