A275329
a(n) = (2+[n/2])*n!/((1+[n/2])*[n/2]!^2).
Original entry on oeis.org
2, 2, 3, 9, 8, 40, 25, 175, 84, 756, 294, 3234, 1056, 13728, 3861, 57915, 14300, 243100, 53482, 1016158, 201552, 4232592, 764218, 17577014, 2912168, 72804200, 11143500, 300874500, 42791040, 1240940160, 164812365, 5109183315, 636438060, 21002455980, 2463251010
Offset: 0
-
a := n -> (2+iquo(n,2))*n!/((1+iquo(n,2))*iquo(n, 2)!^2):
seq(a(n), n=0..34);
-
def A275329():
x, n, k = 1, 1, 2
while True:
yield x * k
if is_odd(n):
x *= n
else:
k += 1
x = (x<<2)//(n+2)
n += 1
a = A275329(); print([next(a) for _ in range(37)])
A329965
a(n) = ((1+n)*floor(1+n/2))*(n!/floor(1+n/2)!)^2.
Original entry on oeis.org
1, 2, 6, 72, 240, 7200, 25200, 1411200, 5080320, 457228800, 1676505600, 221298739200, 821966745600, 149597947699200, 560992303872000, 134638152929280000, 508633022177280000, 155641704786247680000, 591438478187741184000, 224746621711341649920000
Offset: 0
-
A329965 := n -> ((1+n)*floor(1+n/2))*(n!/floor(1+n/2)!)^2:
seq(A329965(n), n=0..19);
-
ser := Series[(1 - Sqrt[1 - 4 x^2] - 4 x^2 (1 - x - Sqrt[1 - 4 x^2]))/(2 x^2 (1 - 4 x^2)^(3/2)), {x, 0, 22}]; Table[n! Coefficient[ser, x, n], {n, 0, 20}]
Table[(1+n)Floor[1+n/2](n!/Floor[1+n/2]!)^2,{n,0,30}] (* Harvey P. Dale, Oct 01 2023 *)
-
from fractions import Fraction
def A329965():
x, n = 1, Fraction(1)
while True:
yield int(x)
m = n if n % 2 else 4/(n+2)
n += 1
x *= m * n
a = A329965(); [next(a) for i in range(36)]
Original entry on oeis.org
1, 5, 14, 27, 34, 11, 62, 183, 98, 39, 142, 107, 220, 69, 44, 495, 322, 143, 436, 139, 482, 637, 574, 119, 674, 233, 782, 55, 898, 29, 1146, 4063, 230, 441, 174, 467, 1516, 493, 662, 239, 1762, 377, 2020, 175, 764, 781, 2302, 2543, 2596, 949, 968, 207, 3126, 1241
Offset: 1
-
f[x_] := GCD[Binomial[x+1, Floor[x+1/2]], Binomial[x, Floor[x/2]]] t=Table[0, {257}]; Do[s=f[n]; If[s<258&&t[[s]]==0, t[[s]]=n], {n, 1, 10000}]; t
A359648
Triangle read by rows. T(n, k) = (n!)^2 / (k! * (n - k)! * (floor(n/2)!)^2 * (floor(n/2) + 1)).
Original entry on oeis.org
1, 1, 1, 1, 2, 1, 3, 9, 9, 3, 2, 8, 12, 8, 2, 10, 50, 100, 100, 50, 10, 5, 30, 75, 100, 75, 30, 5, 35, 245, 735, 1225, 1225, 735, 245, 35, 14, 112, 392, 784, 980, 784, 392, 112, 14, 126, 1134, 4536, 10584, 15876, 15876, 10584, 4536, 1134, 126
Offset: 0
Triangle T(n, k) starts:
[0] 1;
[1] 1, 1;
[2] 1, 2, 1;
[3] 3, 9, 9, 3;
[4] 2, 8, 12, 8, 2;
[5] 10, 50, 100, 100, 50, 10;
[6] 5, 30, 75, 100, 75, 30, 5;
[7] 35, 245, 735, 1225, 1225, 735, 245, 35;
[8] 14, 112, 392, 784, 980, 784, 392, 112, 14;
[9] 126, 1134, 4536, 10584, 15876, 15876, 10584, 4536, 1134, 126;
-
T := proc(n, k) n!^2 / (k! * (n - k)! * iquo(n,2)!^2 * (iquo(n,2) + 1)) end:
for n from 0 to 9 do seq(T(n, k), k = 0..n) od;