A081138 8th binomial transform of (0,0,1,0,0,0, ...).

0, 0, 1, 24, 384, 5120, 61440, 688128, 7340032, 75497472, 754974720, 7381975040, 70866960384, 670014898176, 6253472382976, 57724360458240, 527765581332480, 4785074604081152, 43065671436730368, 385057768140177408

Offset: 0

Author

Paul Barry, Mar 08 2003

Keywords

• easy
• nonn

Comments

Starting at 1, the three-fold convolution of A001018 (powers of 8).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..400
• Index entries for linear recurrences with constant coefficients, signature (24,-192,512).

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), this sequence (q=8), A081139 (q=9), A081140 (q=10), A081141 (q=11), A081142 (q=12), A027476 (q=15).

Programs

Magma

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

Mathematica

LinearRecurrence[{24,-192,512},{0,0,1},30] (* Harvey P. Dale, Jun 08 2014 *)

Formula

a(n) = 24*a(n-1) - 192*a(n-2) + 512*a(n-3) for n>2, a(0)=a(1)=0, a(2)=1.

a(n) = 8^(n-2)*binomial(n, 2).

G.f.: x^2/(1 - 8*x)^3.

E.g.f.: (x^2/2)*exp(8*x). - G. C. Greubel, May 13 2021

From Amiram Eldar, Jan 06 2022: (Start)

Sum_{n>=2} 1/a(n) = 16 - 112*log(8/7).

Sum_{n>=2} (-1)^n/a(n) = 144*log(9/8) - 16. (End)