A379463 a(n) is the total number of paths starting at (0, 0), ending at (n, 0), consisting of steps (1, 1), (1, 0), (1, -3), and staying on or above y = -1.

1, 1, 1, 1, 3, 11, 31, 71, 150, 334, 826, 2146, 5498, 13690, 33762, 84306, 214451, 551107, 1417291, 3637627, 9343555, 24096675, 62439587, 162331747, 422773098, 1102422546, 2879207046, 7534606366, 19756893196, 51894005428, 136496647696, 359478351816, 947912008073

Offset

0,5

Links

Table of n, a(n) for n=0..32.

Formula

a(n) ~ 2^(3/2) * (1 + 4/3^(3/4))^(n + 3/2) / (3^(11/8) * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Jan 15 2025

Conjecture D-finite with recurrence 3*n*(3*n+4)*(n-3)*(3*n+8)*a(n) +3*(-45*n^4+54*n^3+192*n^2-27*n-20)*a(n-1)

+9*(n-1)*(30*n^3-72*n^2-7*n+20)*a(n-2) -3*(n-1)*(n-2)*(90*n^2-234*n+95)*a(n-3) -(n-1)*(n-2)*(n-3)*(121*n+499)*a(n-4) +229*(n-1)*(n-2)*(n-3)*(n-4)*a(n-5)=0. - R. J. Mathar, Jan 29 2025

Example

For n = 4, the a(4)=3 paths are HHHH, UUDU, UUUD, where U=(1,1), D=(1,-3) and H=(1,0).

Maple

A379463 := proc(n)
    add(2*binomial(n, k*4)*binomial(4*k+1, k)/(3*k+2), k=0..floor(n/4)) ;
end proc:
seq(A379463(n), n=0..50) ; # R. J. Mathar, Jan 29 2025

Prog

(PARI) a(n) = sum(k=0, floor(n/4), 2*binomial(n, k*4)*binomial(4*k+1, k)/(3*k+2)) \\ Thomas Scheuerle, Jan 07 2025

Crossrefs

Cf. A069271, A127902, A379464.

Sequence in context: A097081 A093406 A393245 * A376835 A107587 A245931

Adjacent sequences: A379460 A379461 A379462 * A379464 A379465 A379466

Keyword

nonn

Author

Emely Hanna Li Lobnig, Dec 23 2024

Extensions

More terms from Jinyuan Wang, Jan 07 2025

Status

approved