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

0,4

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).

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)

Mathematica

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

Prog

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

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.

Sequence in context: A026006 A024346 A025983 * A020976 A024114 A025982

Adjacent sequences: A081136 A081137 A081138 * A081140 A081141 A081142

Keyword

easy,nonn

Author

Paul Barry, Mar 08 2003

Status

approved