A235535 a(n) = binomial(9*n, 3*n) / (6*n + 1).

1, 12, 1428, 246675, 50067108, 11124755664, 2619631042665, 642312451217745, 162250238001816900, 41932353590942745504, 11034966795189838872624, 2946924270225408943665279, 796607831560617902288322405, 217550867863011281855594752680

Offset

0,2

Comments

This is the case l=6, k=3 of binomial((l+k)*n,k*n)/((l*n+1)/gcd(k,l*n+1)), see Theorem 1.1 in Zhi-Wei Sun's paper.

Also, the sequence follows A002296 and A235536, namely binomial(7*n,n)/(6*n+1) and binomial(8*n,2*n)/(6*n+1); naturally, even binomial(10*n,4*n)/(6*n+1) is always integer.

Links

Robert Israel, Table of n, a(n) for n = 0..403

Zhi-Wei Sun, On Divisibility Of Binomial Coefficients, Journal of the Australian Mathematical Society 93 (2012), p. 189-201.

Formula

a(n) = A001764(3*n) = A047749(6*n).

From Ilya Gutkovskiy, Jun 21 2018: (Start)

G.f.: 6F5(1/9,2/9,4/9,5/9,7/9,8/9; 1/3,1/2,2/3,5/6,7/6; 19683*x/64).

a(n) ~ 3^(9*n-1)/(sqrt(Pi)*4^(3*n+1)*n^(3/2)). (End)

D-finite with recurrence 8*(6*n + 5)*(2*n + 1)*(n + 1)*(3*n + 2)*(3*n + 1)*(6*n + 7)*a(n + 1) = 3*(9*n + 8)*(9*n + 7)*(9*n + 5)*(9*n + 4)*(9*n + 2)*(9*n + 1)*a(n). - Robert Israel, Feb 15 2021

Maple

seq(binomial(9*n, 3*n)/(6*n+1), n=0..30); # Robert Israel, Feb 15 2021

Mathematica

Table[Binomial[9 n, 3 n]/(6 n + 1), {n, 0, 20}]

Prog

(Magma) l:=6; k:=3; [Binomial((l+k)*n, k*n)/(l*n+1): n in [0..20]]; /* here l is divisible by all the prime factors of k */

Crossrefs

Cf. similar sequences generated by binomial((l+k)*n,k*n)/(l*n+1), where l is divisible by all the factors of k: A000108 (l=1, k=1), A001764 (l=2, k=1), A002293 (l=3, k=1), A002294 (l=4, k=1), A002295 (l=5, k=1), A002296 (l=6, k=1), A007556 (l=7, k=1), A062994 (l=8, k=1), A059968 (l=9, k=1), A230388 (l=10, k=1), A048990 (l=2, k=2), A235534 (l=4, k=2), A235536 (l=6, k=2), A187357 (l=3, k=3), this sequence (l=6, k=3).

Sequence in context: A296609 A171484 A230519 * A145835 A008992 A260448

Adjacent sequences: A235532 A235533 A235534 * A235536 A235537 A235538

Keyword

nonn,easy

Author

Bruno Berselli, Jan 12 2014

Status

approved