login
a(n) = 7^(n-2) * C(n,2).
19

%I #44 Sep 08 2022 08:44:49

%S 1,21,294,3430,36015,352947,3294172,29647548,259416045,2219448385,

%T 18643366434,154231485954,1259557135291,10173346092735,81386768741880,

%U 645668365352248,5084638377148953,39779817891812397,309398583602985310

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

%C 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 Vincenzo Librandi, <a href="/A027474/b027474.txt">Table of n, a(n) for n = 2..400</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (21,-147,343).

%F From _Paul Barry_, Mar 08 2003: (Start)

%F G.f.: x^2 / (1-7*x)^3.

%F a(n) = 21*a(n-1) - 147*a(n-2) + 343*a(n-3), a(0) = a(1) = 0, a(2) = 1. (End)

%F 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 E.g.f.: (x^2/2)*exp(7*x). - _G. C. Greubel_, May 13 2021

%F From _Amiram Eldar_, Jan 06 2022: (Start)

%F Sum_{n>=2} 1/a(n) = 14 - 84*log(7/6).

%F Sum_{n>=2} (-1)^n/a(n) = 112*log(8/7) - 14. (End)

%p seq(binomial(n, 2)*7^(n-2), n=2..30); # _Zerinvary Lajos_, Jun 12 2008

%t Table[7^(n-2) Binomial[n,2], {n,2,20}] (* _Harvey P. Dale_, Sep 25 2011 *)

%o (Sage) [7^(n-2)*binomial(n,2) for n in range(2, 21)] # _Zerinvary Lajos_, Mar 13 2009

%o (Magma) [7^(n-2)* Binomial(n, 2): n in [2..20]]; /* _Vincenzo Librandi_, Oct 12 2011 */

%o (PARI) a(n)=7^(n-2)*n*(n-1)/2 \\ _Charles R Greathouse IV_, Oct 07 2015

%Y Third column of A027466.

%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), this sequence (q=7), A081138 (q=8), A081139 (q=9), A081140 (q=10), A081141 (q=11), A081142 (q=12), A027476 (q=15).

%K nonn,easy

%O 2,2

%A _Olivier GĂ©rard_

%E Edited by _Ralf Stephan_, Dec 30 2004