A378850 -id:A378850 - cp's OEIS frontend

A378849 a(n) is the total number of paths starting at (0,0), ending at (n,0), consisting of steps (1,1), (1,0), (1,-2), and staying on or above y = -1.

1, 1, 1, 3, 9, 21, 48, 120, 309, 787, 2011, 5215, 13652, 35894, 94823, 251889, 672285, 1801185, 4842757, 13064059, 35349463, 95912989, 260896318, 711338596, 1943690464, 5321704006, 14597781706, 40112702176, 110404515703, 304338523999, 840140172621, 2322386563353

Offset: 0

Emely Hanna Li Lobnig, Dec 09 2024

nonn

Cf. A071879, A116411, A378850.

a:= proc(n) option remember; `if`(n<4, [1$3, 3][n+1],
      (2*(8*n^3+3*n^2-25*n-6)*a(n-1)-2*(n-1)*(12*n^2-9*n-10)*
       a(n-2)+(43*n+13)*(n-1)*(n-2)*a(n-3)-31*(n-1)*(n-2)*
        (n-3)*a(n-4))/(2*(2*n+3)*(n+3)*(n-2)))
    end:
seq(a(n), n=0..31);  # Alois P. Heinz, Dec 09 2024

a(n) = sum(k=0, floor(n/3), binomial(n, k*3)*binomial(3*k+1, k)/(k+1)) \\ Thomas Scheuerle, Dec 09 2024

a(n) = hypergeom([4/3, (1-n)/3, (2-n)/3, -n/3], [1/3, 3/2, 2], -27/4). - Peter Luschny, Dec 18 2024

More terms from Alois P. Heinz, Dec 09 2024

A379462 a(n) is the total number of paths starting at (0, 0), ending at (n, 0), consisting of steps (1, 1), (1, 0), (1, -2), and staying on or above y = -3.

Original entry on oeis.org

1, 1, 1, 4, 13, 31, 75, 204, 561, 1499, 4001, 10814, 29364, 79704, 216672, 590764, 1614421, 4419049, 12116139, 33277722, 91546143, 252209535, 695803659, 1922166420, 5316714156, 14723570406, 40820144106, 113293243636, 314759548879, 875342190283, 2436582442381

Offset: 0

Emely Hanna Li Lobnig, Dec 23 2024

nonn

			For n = 3, the a(3)=4 paths are DUU, HHH, UDU, UUD, where U=(1,1), D=(1,-2) and H=(1,0). An example of a path with these steps, but not staying on or above y = -3, is for n=6: DDUUUU.

Cf. A071879, A116411, A378849, A378850.

lista(nn) = my(v=vector(nn+5), w); print1(v[4]=1); for(n=1, nn, w=v; for(i=1, n+3, w[i]+=v[i+2]; w[i+1]+=v[i]); v=w; print1(", ", v[4])); \\ Jinyuan Wang, Jan 07 2025

a(n) = Sum_{k=0..floor(n/3)} 2*binomial(n, k*3)*(binomial(3*k+3, k)/(k+2) - binomial(3*k, k-1)/(k+1)). - Thomas Scheuerle, Jan 07 2025

a(n) ~ 23 * (1 + 3/2^(2/3))^(n + 3/2) / (4 * sqrt(3*Pi) * n^(3/2)). - Vaclav Kotesovec, Jan 15 2025

More terms from Jinyuan Wang, Jan 07 2025

cp's OEIS Frontend

A378849 a(n) is the total number of paths starting at (0,0), ending at (n,0), consisting of steps (1,1), (1,0), (1,-2), and staying on or above y = -1.

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