A081140 10th binomial transform of (0,0,1,0,0,0,...).

0, 0, 1, 30, 600, 10000, 150000, 2100000, 28000000, 360000000, 4500000000, 55000000000, 660000000000, 7800000000000, 91000000000000, 1050000000000000, 12000000000000000, 136000000000000000, 1530000000000000000

Offset: 0

Author

Paul Barry, Mar 08 2003

Keywords

• easy
• nonn

Comments

Starting at 1, the three-fold convolution of A011557 (powers of 10).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..400
• Index entries for linear recurrences with constant coefficients, signature (30,-300,1000).

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

Programs

Magma

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

Mathematica

Table[10^(n-2)*Binomial[n, 2], {n, 0, 30}] (* G. C. Greubel, May 13 2021 *)

Formula

a(n) = 30*a(n-1) - 300*a(n-2) + 1000*a(n-3), a(0)=a(1)=0, a(2)=1.

a(n) = 10^(n-2)*binomial(n, 2).

G.f.: x^2/(1-10*x)^3.

E.g.f.: (x^2/2)*exp(10*x). - G. C. Greubel, May 13 2021

From Amiram Eldar, Jan 06 2022: (Start)

Sum_{n>=2} 1/a(n) = 20 - 180*log(10/9).

Sum_{n>=2} (-1)^n/a(n) = 220*log(11/10) - 20. (End)