A169958 a(n) = binomial(9*n, n).

1, 9, 153, 2925, 58905, 1221759, 25827165, 553270671, 11969016345, 260887834350, 5720645481903, 126050526132804, 2788629694000605, 61902409203193230, 1378095785451705375, 30756373941461374800, 687917389635036844569, 15415916972482007401455, 346051021610256116115150

Offset: 0

Author

N. J. A. Sloane, Aug 07 2010

Keywords

• nonn

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..110

Crossrefs

binomial(k*n,n): A000984 (k = 2), A005809 (k = 3), A005810 (k = 4), A001449 (k = 5), A004355 (k = 6), A004368 (k = 7), A004381 (k = 8), A169959 - A169961 (k = 10 thru 12).

Programs

Magma

[Binomial(9*n, n): n in [0..50] ]; // Vincenzo Librandi, Apr 21 2011

Mathematica

A169958[n_] := Binomial[9*n, n]; Array[A169958, 20, 0] (* Paolo Xausa, Aug 20 2025 *)

Formula

a(n) = C(9*n-1, n-1)*C(81*n^2, 2)/(3*n*C(9*n+1, 3)), n > 0. - Gary Detlefs, Jan 02 2014

From Peter Bala, Feb 21 2022: (Start)

The o.g.f. A(x) is algebraic: (1 - A(x))*(1 + 8*A(x))^8 + (9^9)*x*A(x)^9 = 0.

Sum_{n >= 1} a(n)*( x*(8*x + 9)^8/(9^9*(1 + x)^9) )^n = x. (End)

D-finite with recurrence 128*n*(8*n-5) *(4*n-1) *(8*n-7) *(2*n-1) *(8*n-1) *(4*n-3) *(8*n-3)*a(n) -81*(9*n-7) *(9*n-5) *(3*n-1) *(9*n-1) *(9*n-8) *(3*n-2) *(9*n-4) *(9*n-2)*a(n-1)=0. - R. J. Mathar, Aug 19 2025

G.f.: 8F7(8/9, 7/9, 2/3, 5/9, 4/9, 1/3, 2/9 ,1/9 ; 7/8, 3/4, 5/8, 1/2, 3/8, 1/4, 1/8; 387420489/16777216*x). - R. J. Mathar, Aug 19 2025