A274117 a(n) = ((6n-5)!!!+(6n-4)!!!)/(6n-3).

1, 12, 1064, 252160, 115315200, 86449126400, 96313245952000, 149342026677043200, 307513455044956160000, 811744577542368870400000, 2672529840751688498380800000, 10735527449319396895332761600000, 51677469466519591978527317032960000, 293652804750537765304678163152896000000

Offset

1,2

Comments

Sequence is similar to A273889, with a similar proof of divisibility.

Links

Brian Cheung, Table of n, a(n) for n = 1..100

Formula

a(n) = ((6n-5)!!!+(6n-4)!!!)/(6n-3).

D-finite with recurrence: 4*(6*n + 5)*(3*n + 2)*(2*n + 1)*(3*n + 4)*(6*n + 7)*(n + 2)*a(n + 1) - 2*(18*n^2 + 54*n + 35)*(2*n + 3)^2*a(n + 2) + (2*n + 5)*(n + 1)*a(n + 3) = 0. - Robert Israel, Mar 08 2026

Example

a(1) = (1+2)/3 = 1;
a(2) = (1*4*7+2*5*8)/9 = 12;
a(3) = (1*4*7*10*13+2*5*8*11*14)/15 = 1064.

Maple

tf:= proc(n) option remember; n * tf(n-3) end proc:
tf(1):= 1; tf(2):= 2:
f:= n -> (tf(6*n-5) + tf(6*n-4))/(6*n-3):
map(f, [$1..50]); # Robert Israel, Mar 08 2026

Mathematica

B[n_, k_] := (Product[k (i - 1) + 1, {i, 2 n - 1}] + Product[k (i - 1) + 2, {i, 2 n - 1}])/(2 k (n - 1) + 3); Table[B[n, 3], {n, 14}] (* Michael De Vlieger, Jun 10 2016 *)
trp[n_]:=Times@@Range[n, 1, -3]; Table[(trp[6n-5]+trp[6n-4])/(6n-3), {n, 15}] (* Harvey P. Dale, Feb 08 2026 *)

Prog

(Python)
triplefac=lambda x:1 if x<2 else x*triplefac(x-3)
for i in range(1, 101):
    print(i, (triplefac(6*i-5)+triplefac(6*i-4))//(6*i-3))

Crossrefs

Cf. A007661, A273889.

Sequence in context: A232912 A004812 A088671 * A295427 A354822 A260030

Adjacent sequences: A274114 A274115 A274116 * A274118 A274119 A274120

Keyword

nonn

Author

Hong-Chang Wang and Brian Cheung, Jun 10 2016

Status

approved