This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A017389 #51 May 04 2025 15:10:13
%S A017389 1,15,154,1350,10891,83685,623764,4558380,32875381,234980955,
%T A017389 1669192174,11806149810,83252603071,585817587825,4115974729384,
%U A017389 28888095527640,202598073849961,1420093671872295,9950191865139394,69699025028403870,488131588547752051,3418113197039242365
%N A017389 Expansion of g.f. 1/((1-3*x)*(1-5*x)*(1-7*x)).
%H A017389 Vincenzo Librandi, <a href="/A017389/b017389.txt">Table of n, a(n) for n = 0..200</a>
%H A017389 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (15,-71,105).
%F A017389 From _Vincenzo Librandi_, Jun 26 2013: (Start)
%F A017389 a(n) = 15*a(n-1) - 71*a(n-2) + 105*a(n-3).
%F A017389 a(n) = 12*a(n-1) - 35*a(n-2) + 3^n. (End)
%F A017389 a(n) = (7^(n+2) - 2*5^(n+2) + 3^(n+2))/8. - _Yahia Kahloune_, Aug 13 2013
%F A017389 From _Seiichi Manyama_, May 04 2025: (Start)
%F A017389 a(n) = Sum_{k=0..n} 2^k * 3^(n-k) * binomial(n+2,k+2) * Stirling2(k+2,2).
%F A017389 a(n) = Sum_{k=0..n} (-2)^k * 7^(n-k) * binomial(n+2,k+2) * Stirling2(k+2,2). (End)
%F A017389 E.g.f.: exp(3*x)*(9 - 50*exp(2*x) + 49*exp(4*x))/8. - _Stefano Spezia_, May 04 2025
%p A017389 A017389:=n->(7^(n+2) - 2*5^(n+2) + 3^(n+2))/8: seq(A017389(n), n=0..20); # _Wesley Ivan Hurt_, Mar 25 2014
%t A017389 CoefficientList[Series[1 / ((1 - 3 x) (1 - 5 x) (1 - 7 x)), {x, 0, 20}], x] (* _Vincenzo Librandi_, Jun 26 2013 *)
%o A017389 (Magma) I:=[1,15,154]; [n le 3 select I[n] else 15*Self(n-1)-71*Self(n-2)+105*Self(n-3): n in [1..20]]; // _Vincenzo Librandi_, Jun 26 2013
%o A017389 (Magma) m:=20; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/((1-3*x)*(1-5*x)*(1-7*x)))); // _Vincenzo Librandi_, Jun 26 2013
%o A017389 (PARI) a(n) = (7^(n+2) - 2*5^(n+2) + 3^(n+2))/8; \\ _Joerg Arndt_, Aug 13 2013
%o A017389 (PARI) x='x+O('x^20); Vec(1/((1-3*x)*(1-5*x)*(1-7*x))) \\ _Altug Alkan_, Sep 23 2018
%Y A017389 Cf. A019333.
%K A017389 nonn,easy
%O A017389 0,2
%A A017389 _N. J. A. Sloane_