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A027474 a(n) = 7^(n-2) * C(n,2).

%I A027474 #44 Sep 08 2022 08:44:49
%S A027474 1,21,294,3430,36015,352947,3294172,29647548,259416045,2219448385,
%T A027474 18643366434,154231485954,1259557135291,10173346092735,81386768741880,
%U A027474 645668365352248,5084638377148953,39779817891812397,309398583602985310
%N A027474 a(n) = 7^(n-2) * C(n,2).
%C A027474 7th binomial transform of (0,0,1,0,0,0,........). Starting at 1, the three-fold convolution of A000420 (powers of 7). - _Paul Barry_, Mar 08 2003
%H A027474 Vincenzo Librandi, <a href="/A027474/b027474.txt">Table of n, a(n) for n = 2..400</a>
%H A027474 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (21,-147,343).
%F A027474 From _Paul Barry_, Mar 08 2003: (Start)
%F A027474 G.f.: x^2 / (1-7*x)^3.
%F A027474 a(n) = 21*a(n-1) - 147*a(n-2) + 343*a(n-3), a(0) = a(1) = 0, a(2) = 1. (End)
%F A027474 Numerators of sequence a[3,n] in (a[i,j])^3 where a[i,j] = binomial(i-1, j-1)/2^(i-1) if j<=i, 0 if j>i.
%F A027474 E.g.f.: (x^2/2)*exp(7*x). - _G. C. Greubel_, May 13 2021
%F A027474 From _Amiram Eldar_, Jan 06 2022: (Start)
%F A027474 Sum_{n>=2} 1/a(n) = 14 - 84*log(7/6).
%F A027474 Sum_{n>=2} (-1)^n/a(n) = 112*log(8/7) - 14. (End)
%p A027474 seq(binomial(n, 2)*7^(n-2), n=2..30); # _Zerinvary Lajos_, Jun 12 2008
%t A027474 Table[7^(n-2) Binomial[n,2], {n,2,20}] (* _Harvey P. Dale_, Sep 25 2011 *)
%o A027474 (Sage) [7^(n-2)*binomial(n,2) for n in range(2, 21)] # _Zerinvary Lajos_, Mar 13 2009
%o A027474 (Magma) [7^(n-2)* Binomial(n, 2): n in [2..20]]; /* _Vincenzo Librandi_, Oct 12 2011 */
%o A027474 (PARI) a(n)=7^(n-2)*n*(n-1)/2 \\ _Charles R Greathouse IV_, Oct 07 2015
%Y A027474 Third column of A027466.
%Y A027474 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), this sequence (q=7), A081138 (q=8), A081139 (q=9), A081140 (q=10), A081141 (q=11), A081142 (q=12), A027476 (q=15).
%K A027474 nonn,easy
%O A027474 2,2
%A A027474 _Olivier Gérard_
%E A027474 Edited by _Ralf Stephan_, Dec 30 2004