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%I #36 Sep 08 2022 08:45:09
%S 0,0,1,27,486,7290,98415,1240029,14880348,172186884,1937102445,
%T 21308126895,230127770466,2447722649502,25701087819771,
%U 266895911974545,2745215094595320,28001193964872264,283512088894331673
%N 9th binomial transform of (0,0,1,0,0,0,...).
%C Starting at 1, the three-fold convolution of A001019 (powers of 9).
%H Vincenzo Librandi, <a href="/A081139/b081139.txt">Table of n, a(n) for n = 0..400</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (27,-243,729).
%F a(n) = 27*a(n-1) - 243*a(n-2) + 729*a(n-3), a(0)=a(1)=0, a(2)=1.
%F a(n) = 9^(n-2)*binomial(n, 2).
%F G.f.: x^2/(1-9*x)^3.
%F E.g.f.: (x^2/2)*exp(9*x). - _G. C. Greubel_, May 13 2021
%F From _Amiram Eldar_, Jan 06 2022: (Start)
%F Sum_{n>=2} 1/a(n) = 18 - 144*log(9/8).
%F Sum_{n>=2} (-1)^n/a(n) = 180*log(10/9) - 18. (End)
%t LinearRecurrence[{27,-243,729},{0,0,1},30] (* _Harvey P. Dale_, Jan 30 2018 *)
%o (Magma) [9^n* Binomial(n+2, 2): n in [-2..20]]; // _Vincenzo Librandi_, Oct 16 2011
%Y 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).
%Y Cf. A001019.
%K easy,nonn
%O 0,4
%A _Paul Barry_, Mar 08 2003
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