|
Search: id:A101974
|
|
|
| A101974 |
|
Triangle read by rows: number of Dyck paths of semilength n with k peaks before the first return (1<= k <n). |
|
+0
2
|
|
| 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
(list; graph; listen)
|
|
|
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).
|
|
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: 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
|
Row sums are the Catalan numbers (A000108). Column k=1 yields the partial sums of the Catalan sequence (A014137).
Cf. A000108, A014137, A101975.
Sequence in context: A019823 A092107 A114489 this_sequence A097607 A132893 A091958
Adjacent sequences: A101971 A101972 A101973 this_sequence A101975 A101976 A101977
|
|
KEYWORD
|
nonn,tabf
|
|
AUTHOR
|
Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 22 2004
|
|
|
Search completed in 0.002 seconds
|
