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

%I A081140 #33 Sep 08 2022 08:45:09
%S A081140 0,0,1,30,600,10000,150000,2100000,28000000,360000000,4500000000,
%T A081140 55000000000,660000000000,7800000000000,91000000000000,
%U A081140 1050000000000000,12000000000000000,136000000000000000,1530000000000000000
%N A081140 10th binomial transform of (0,0,1,0,0,0,...).
%C A081140 Starting at 1, the three-fold convolution of A011557 (powers of 10).
%H A081140 Vincenzo Librandi, <a href="/A081140/b081140.txt">Table of n, a(n) for n = 0..400</a>
%H A081140 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (30,-300,1000).
%F A081140 a(n) = 30*a(n-1) - 300*a(n-2) + 1000*a(n-3), a(0)=a(1)=0, a(2)=1.
%F A081140 a(n) = 10^(n-2)*binomial(n, 2).
%F A081140 G.f.: x^2/(1-10*x)^3.
%F A081140 E.g.f.: (x^2/2)*exp(10*x). - _G. C. Greubel_, May 13 2021
%F A081140 From _Amiram Eldar_, Jan 06 2022: (Start)
%F A081140 Sum_{n>=2} 1/a(n) = 20 - 180*log(10/9).
%F A081140 Sum_{n>=2} (-1)^n/a(n) = 220*log(11/10) - 20. (End)
%t A081140 Table[10^(n-2)*Binomial[n, 2], {n, 0, 30}] (* _G. C. Greubel_, May 13 2021 *)
%o A081140 (Magma) [10^n* Binomial(n+2, 2): n in [-2..20]]; // _Vincenzo Librandi_, Oct 16 2011
%Y A081140 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).
%K A081140 easy,nonn
%O A081140 0,4
%A A081140 _Paul Barry_, Mar 08 2003