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A081141 11th binomial transform of (0,0,1,0,0,0,...).

%I A081141 #31 Sep 08 2022 08:45:09
%S A081141 0,0,1,33,726,13310,219615,3382071,49603708,701538156,9646149645,
%T A081141 129687123005,1711870023666,22254310307658,285596982281611,
%U A081141 3624884775112755,45569980029988920,568105751040528536
%N A081141 11th binomial transform of (0,0,1,0,0,0,...).
%C A081141 Starting at 1, the three-fold convolution of A001020 (powers of 11).
%H A081141 Vincenzo Librandi, <a href="/A081141/b081141.txt">Table of n, a(n) for n = 0..400</a>
%H A081141 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (33,-363,1331).
%F A081141 a(n) = 33*a(n-1) - 363*a(n-2) + 1331*a(n-3), a(0) = a(1) = 0, a(2) = 1.
%F A081141 a(n) = 11^(n-2)*binomial(n, 2).
%F A081141 G.f.: x^2/(1 - 11*x)^3.
%F A081141 E.g.f.: (1/2)*exp(11*x)*x^2. - _Franck Maminirina Ramaharo_, Nov 23 2018
%F A081141 From _Amiram Eldar_, Jan 06 2022: (Start)
%F A081141 Sum_{n>=2} 1/a(n) = 22 - 220*log(11/10).
%F A081141 Sum_{n>=2} (-1)^n/a(n) = 264*log(12/11) - 22. (End)
%p A081141 seq((11)^(n-2)*binomial(n,2), n=0..30); # _G. C. Greubel_, May 13 2021
%t A081141 LinearRecurrence[{33,-363,1331},{0,0,1},30] (* _Harvey P. Dale_, Dec 15 2014 *)
%o A081141 (Magma) [11^(n-2)*Binomial(n, 2): n in [0..20]]; // _Vincenzo Librandi_, Oct 16 2011
%o A081141 (PARI) vector(20, n, n--; 11^(n-2)*binomial(n, 2)) \\ _G. C. Greubel_, Nov 23 2018
%o A081141 (Sage) [11^(n-2)*binomial(n, 2) for n in range(20)] # _G. C. Greubel_, Nov 23 2018
%Y A081141 Cf. A001020.
%Y A081141 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), A081140 (q=10), this sequence (q=11), A081142 (q=12), A027476 (q=15).
%K A081141 easy,nonn
%O A081141 0,4
%A A081141 _Paul Barry_, Mar 08 2003