A374835 Number of ternary paths of length 3*n having exactly 1 hill.

0, 1, 0, 4, 14, 72, 370, 1995, 11064, 62774, 362614, 2125479, 12610410, 75584363, 457000668, 2783991972, 17071362986, 105287143740, 652687596388, 4064618917998, 25416557913498, 159523021737403, 1004594882114020, 6345859437931884, 40198433775513524

Offset

0,4

Links

Table of n, a(n) for n=0..24.

Naiomi Cameron and J. E. McLeod, Returns and Hills on Generalized Dyck Paths, Journal of Integer Sequences, Vol. 19, 2016, #16.6.1.

Formula

a(n) = Sum_{k=0..n-1} (-1)^k * (k+1)*(k+2)/(2*n-k) * binomial(3*n-2-2*k,n-1-k).

G.f.: x * (g/(1 + x*g))^2, where g = 1 + x*g^3.

Prog

(PARI) a(n) = sum(k=0, n-1, (-1)^k*(k+1)*(k+2)/(2*n-k)*binomial(3*n-2-2*k, n-1-k));

Crossrefs

Column k=1 of A101371.

Cf. A001764, A023053, A065601.

Sequence in context: A322700 A167144 A159483 * A088127 A242739 A007282

Adjacent sequences: A374832 A374833 A374834 * A374836 A374837 A374838

Keyword

nonn

Author

Seiichi Manyama, Jul 21 2024

Status

approved