A302182
Number of 3D walks of type abc.
Original entry on oeis.org
1, 1, 5, 12, 62, 200, 1065, 3990, 21714, 89082, 492366, 2147376, 12004740, 54718092, 308559537, 1454116950, 8255788970, 39935276810, 227976044010, 1126178350440, 6457854821340, 32456552441040, 186814834574550, 952569927106980, 5500292590186380, 28391993275117500
Offset: 0
Cf.
A000108,
A000984,
A002212,
A002896,
A005572,
A026375,
A064037,
A081671,
A138547,
A145847,
A145867,
A150500,
A202814.
-
from math import comb as binomial
def row(n: int) -> list[int]:
return sum(binomial(n, k)*binomial(k, k//2)//(k//2+1)*((k+1) %2)*binomial(n-k, (n-k)//2)**2 for k in range(n+1))
for n in range(26): print(row(n)) # Mélika Tebni, Nov 27 2024
A302185
Number of 3D n-step walks of type acc.
Original entry on oeis.org
1, 2, 7, 24, 98, 400, 1785, 7980, 37674, 178164, 874146, 4294752, 21667932, 109436184, 563910633, 2908233900, 15235550330, 79870553620, 424021948350, 2252356700880, 12088746573540, 64913104882080, 351594254659830, 1905139854213960, 10399223643879420, 56783986550235000
Offset: 0
Cf.
A000108,
A000984,
A002212,
A002896,
A005572,
A026375,
A064037,
A081671,
A138547,
A145847,
A145867,
A150500,
A202814.
-
b:= n-> binomial(n, floor(n/2))*binomial(n+1, floor((n+1)/2)):
C:= n-> binomial(2*n, n)/(n+1):
a:= n-> add(binomial(n, 2*k)*C(k)*b(n-2*k), k=0..n/2):
seq(a(n), n=0..25); # Alois P. Heinz, Dec 06 2024
# second Maple program:
a:= proc(n) option remember; `if`(n<4, [1, 2, 7, 24][n+1],
(8*(14*n^4+85*n^3+190*n^2+188*n+63)*a(n-1)+4*(n-1)*
(80*n^4+418*n^3+676*n^2+269*n-108)*a(n-2)-96*(n-1)*(n-2)*
(10*n^2+31*n+27)*a(n-3)-144*(n-1)*(n-2)*(n-3)*(8*n^2+33*n+36)*
a(n-4))/((n+4)*(n+3)*(n+2)*(8*n^2+17*n+11)))
end:
seq(a(n), n=0..25); # Alois P. Heinz, Dec 06 2024
-
b[n_] := Binomial[n, Floor[n/2]]*Binomial[n+1, Floor[(n+1)/2]];
c[n_] := Binomial[2*n, n]/(n+1);
a[n_] := Sum[Binomial[n, 2*k]*c[k]*b[n - 2*k], {k, 0, n/2}];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Jan 28 2025, after Alois P. Heinz *)
-
from math import comb as binomial
def C(n): return (binomial(2*n, n)//(n+1)) # Catalan numbers
def a(n):
return sum(binomial(n, k)*C((k+1)//2)*C(k//2)*(2*(k//2)+1)*binomial(n-k, (n-k)//2) for k in range(n+1))
print([a(n) for n in range(26)]) # Mélika Tebni, Dec 06 2024
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