A081139 9th binomial transform of (0,0,1,0,0,0,...).

0, 0, 1, 27, 486, 7290, 98415, 1240029, 14880348, 172186884, 1937102445, 21308126895, 230127770466, 2447722649502, 25701087819771, 266895911974545, 2745215094595320, 28001193964872264, 283512088894331673

Offset: 0

Author

Paul Barry, Mar 08 2003

Keywords

• easy
• nonn

Comments

Starting at 1, the three-fold convolution of A001019 (powers of 9).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..400
• Index entries for linear recurrences with constant coefficients, signature (27,-243,729).

Crossrefs

Sequences similar to the form q^(n-2)*binomial(n, 2): A000217 (q=1), A001788 (q=2), A027472 (q=3), A038845 (q=4), A081135 (q=5), A081136 (q=6), A027474 (q=7), A081138 (q=8), this sequence (q=9), A081140 (q=10), A081141 (q=11), A081142 (q=12), A027476 (q=15).

Cf. A001019.

Programs

Magma

[9^n* Binomial(n+2, 2): n in [-2..20]]; // Vincenzo Librandi, Oct 16 2011

Mathematica

LinearRecurrence[{27,-243,729},{0,0,1},30] (* Harvey P. Dale, Jan 30 2018 *)

Formula

a(n) = 27*a(n-1) - 243*a(n-2) + 729*a(n-3), a(0)=a(1)=0, a(2)=1.

a(n) = 9^(n-2)*binomial(n, 2).

G.f.: x^2/(1-9*x)^3.

E.g.f.: (x^2/2)*exp(9*x). - G. C. Greubel, May 13 2021

From Amiram Eldar, Jan 06 2022: (Start)

Sum_{n>=2} 1/a(n) = 18 - 144*log(9/8).

Sum_{n>=2} (-1)^n/a(n) = 180*log(10/9) - 18. (End)