A374836 Number of ternary paths of length 3*n having exactly 2 hills.

0, 0, 1, 0, 6, 21, 114, 597, 3278, 18420, 105618, 615331, 3632352, 21678975, 130598887, 793085742, 4849791942, 29838388707, 184573315170, 1147219365762, 7161284719412, 44876343104655, 282206695488603, 1780341161583746, 11264338644455334, 71461150894269030

Offset

0,5

Links

Table of n, a(n) for n=0..25.

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-2} (-1)^k * (k+1)*(k+2)*(k+3)/(2*(2*n-1-k)) * binomial(3*n-4-2*k,n-2-k).

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

Prog

(PARI) a(n) = sum(k=0, n-2, (-1)^k*(k+1)*(k+2)*(k+3)/(2*(2*n-1-k))*binomial(3*n-4-2*k, n-2-k));

Crossrefs

Column k=2 of A101371.

Cf. A001764, A023053, A294527.

Sequence in context: A364328 A298837 A009253 * A365880 A251593 A012840

Adjacent sequences: A374833 A374834 A374835 * A374837 A374838 A374839

Keyword

nonn

Author

Seiichi Manyama, Jul 21 2024

Status

approved