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A027472 Third convolution of the powers of 3 (A000244).

%I A027472 #60 Sep 08 2022 08:44:49
%S A027472 1,9,54,270,1215,5103,20412,78732,295245,1082565,3897234,13817466,
%T A027472 48361131,167403915,573956280,1951451352,6586148313,22082967873,
%U A027472 73609892910,244074908070,805447196631,2646469360359,8661172452084,28242953648100,91789599356325,297398301914493,960825283108362,3095992578904722
%N A027472 Third convolution of the powers of 3 (A000244).
%C A027472 Third column of A027465.
%C A027472 With offset = 2, a(n) is the number of length n words on alphabet {u,v,w,z} such that each word contains exactly 2 u's. - _Zerinvary Lajos_, Dec 29 2007
%H A027472 G. C. Greubel, <a href="/A027472/b027472.txt">Table of n, a(n) for n = 3..1000</a>
%H A027472 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (9,-27,27).
%F A027472 Numerators of sequence a[3,n] in (b^2)[i,j]) where b[i,j] = binomial(i-1, j-1)/2^(i-1) if j <= i, 0 if j > i.
%F A027472 From _Wolfdieter Lang_: (Start)
%F A027472 a(n) = 3^(n-3)*binomial(n-1, 2).
%F A027472 G.f.: (x/(1-3*x))^3. (Third convolution of A000244, powers of 3.) (End)
%F A027472 a(n) = |A075513(n, 2)|/9, n >= 3.
%F A027472 a(n) = A152818(n-3,2)/2 = A006043(n-3)/2. - _Paul Curtz_, Jan 07 2009
%F A027472 The sequence 0, 1, 9, 54, ... has e.g.f.: (x + 3*x^2/2)*exp(3*x)/. - _Paul Barry_, Jul 23 2003
%F A027472 E.g.f.: E(0) where E(k) = 1 + 3*(2*k+3)*x/((2*k+1)^2 - 3*x*(k+2)*(2*k+1)^2/(3*x*(k+2) + 2*(k+1)^2/E(k+1))); (continued fraction, 3-step). - _Sergei N. Gladkovskii_, Nov 23 2012
%F A027472 With offset=2 e.g.f.: x^2*exp(3*x)/2. - _Geoffrey Critzer_, Oct 03 2013
%F A027472 From _Amiram Eldar_, Jan 05 2022: (Start)
%F A027472 Sum_{n>=3} 1/a(n) = 6 - 12*log(3/2).
%F A027472 Sum_{n>=3} (-1)^(n+1)/a(n) = 24*log(4/3) - 6. (End)
%t A027472 nn=41; Drop[Range[0,nn]!CoefficientList[Series[Exp[x]^3 x^2/2!,{x,0,nn}],x],2] (* _Geoffrey Critzer_, Oct 03 2013 *)
%t A027472 LinearRecurrence[{9,-27,27}, {1,9,54}, 40] (* _G. C. Greubel_, May 12 2021 *)
%t A027472 Abs[Take[CoefficientList[Series[1/(1+3x^2)^3,{x,0,60}],x],{1,-1,2}]] (* _Harvey P. Dale_, Mar 03 2022 *)
%o A027472 (Sage) [3^(n-3)*binomial(n-1,2) for n in range(3, 40)] # _Zerinvary Lajos_, Mar 10 2009
%o A027472 (PARI) a(n)=([0,1,0; 0,0,1; 27,-27,9]^(n-3)*[1;9;54])[1,1] \\ _Charles R Greathouse IV_, Oct 03 2016
%o A027472 (Magma) [3^(n-3)*Binomial(n-1, 2): n in [3..40]]; // _G. C. Greubel_, May 12 2021
%Y A027472 Cf. A000244, A006043, A027465, A075513, A152818.
%Y A027472 Sequences similar to the form q^(n-2)*binomial(n, 2): A000217 (q=1), A001788 (q=2), this sequence (q=3), A038845 (q=4), A081135 (q=5), A081136 (q=6), A027474 (q=7), A081138 (q=8), A081139 (q=9), A081140 (q=10), A081141 (q=11), A081142 (q=12), A027476 (q=15).
%K A027472 nonn,easy
%O A027472 3,2
%A A027472 _Olivier Gérard_
%E A027472 Corrected by _T. D. Noe_, Nov 07 2006
%E A027472 Better name from _Wolfdieter Lang_
%E A027472 Terms a(23) onward added by _G. C. Greubel_, May 12 2021