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Third column of A027467.
13

%I #22 Sep 08 2022 08:44:49

%S 1,45,1350,33750,759375,15946875,318937500,6150937500,115330078125,

%T 2114384765625,38058925781250,674680957031250,11806916748046875,

%U 204350482177734375,3503151123046875000,59553569091796875000

%N Third column of A027467.

%H Vincenzo Librandi, <a href="/A027476/b027476.txt">Table of n, a(n) for n = 3..200</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (45,-675,3375).

%F Numerators of sequence a[3,n] in (a[i,j])^4 where a[i,j] = binomial(i-1, j-1)/2^(i-1) if j<=i, 0 if j>i.

%F a(n) = 15^(n-3)*binomial(n-1, 2).

%F From _G. C. Greubel_, May 13 2021: (Start)

%F a(n) = 45*a(n-1) - 675*a(n-2) + 3375*a(n-3).

%F G.f.: x^3/(1 - 15*x)^3.

%F E.g.f.: (-2 + (2 - 30*x + 225*x^2)*exp(15*x))/6750. (End)

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

%F Sum_{n>=3} 1/a(n) = 30 - 420*log(15/14).

%F Sum_{n>=3} (-1)^(n+1)/a(n) = 480*log(16/15) - 30. (End)

%p seq((15)^(n-3)*binomial(n-1, 2), n=3..30) # _G. C. Greubel_, May 13 2021

%t Table[(n-1)*(n-2)/2 * 15^(n-3), {n, 3, 30}] (* _Vincenzo Librandi_, Dec 29 2012 *)

%o (Magma) [(n-1)*(n-2)/2 * 15^(n-3): n in [3..20]]; // _Vincenzo Librandi_, Dec 29 2012

%o (Sage) [(15)^(n-3)*binomial(n-1,2) for n in (3..30)] # _G. C. Greubel_, May 13 2021

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

%K nonn

%O 3,2

%A _Olivier GĂ©rard_