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 A019943 #31 May 04 2025 14:37:49
%S A019943 1,21,298,3570,38971,401751,3988468,38583300,366449941,3434404281,
%T A019943 31873887838,293663563830,2690806228111,24553315831611,
%U A019943 223338364450408,2026585451393160,18355202849805481,166009125098571741,1499772036736668178,13537796780062999290
%N A019943 Expansion of 1/((1-5*x)*(1-7*x)*(1-9*x)).
%H A019943 Vincenzo Librandi, <a href="/A019943/b019943.txt">Table of n, a(n) for n = 0..200</a>
%H A019943 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (21,-143,315).
%F A019943 a(n) = (5^(n+2) - 2*7^(n+2) + 9^(n+2))/8. - _R. J. Mathar_, Jun 29 2013
%F A019943 a(0)=1, a(1)=21, a(2)=298; for n>2, a(n) = 21*a(n-1) -143*a(n-2) +315*a(n-3). - _Vincenzo Librandi_, Jul 03 2013
%F A019943 a(n) = 16*a(n-1) -63*a(n-2) +5^n. - _Vincenzo Librandi_, Jul 03 2013
%F A019943 E.g.f.: (25*exp(5*x) - 98*exp(7*x) + 81*exp(9*x))/8. - _G. C. Greubel_, Nov 24 2018
%F A019943 From _Seiichi Manyama_, May 04 2025: (Start)
%F A019943 a(n) = Sum_{k=0..n} 2^k * 5^(n-k) * binomial(n+2,k+2) * Stirling2(k+2,2).
%F A019943 a(n) = Sum_{k=0..n} (-2)^k * 9^(n-k) * binomial(n+2,k+2) * Stirling2(k+2,2). (End)
%p A019943 a:= n-> (Matrix(3, (i, j)-> `if`(i=j-1, 1, `if`(j=1, [21, -143, 315][i], 0)))^n)[1, 1]: seq(a(n), n=0..25); # _Alois P. Heinz_, Jul 03 2013
%t A019943 CoefficientList[Series[1/((1-5x)(1-7x)(1-9 x)), {x, 0, 20}], x] (* _Vincenzo Librandi_, Jul 03 2013 *)
%t A019943 LinearRecurrence[{21,-143,315}, {1, 21, 298}, 20] (* or *) Table[(5^(n+2) - 2*7^(n+2) + 9^(n+2))/8, {n,0,20}] (* _G. C. Greubel_, Nov 24 2018 *)
%o A019943 (Magma) m:=20; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/((1-5*x)*(1-7*x)*(1-9*x)))); // _Vincenzo Librandi_, Jul 03 2013
%o A019943 (Magma) I:=[1, 21, 298]; [n le 3 select I[n] else 21*Self(n-1)-143*Self(n-2)+315*Self(n-3): n in [1..20]]; // _Vincenzo Librandi_, Jul 03 2013
%o A019943 (Magma) [(5^(n+2) - 2*7^(n+2) + 9^(n+2))/8: n in [0..20]]; // _G. C. Greubel_, Nov 24 2018
%o A019943 (PARI) vector(20, n, n--; (5^(n+2) - 2*7^(n+2) + 9^(n+2))/8) \\ _G. C. Greubel_, Nov 24 2018
%o A019943 (Sage) [(5^(n+2) - 2*7^(n+2) + 9^(n+2))/8 for n in range(20)] # _G. C. Greubel_, Nov 24 2018
%o A019943 (GAP) List([0..20], n -> (5^(n+2) - 2*7^(n+2) + 9^(n+2))/8); # _G. C. Greubel_, Nov 24 2018
%Y A019943 Cf. A017389, A019333.
%K A019943 nonn,easy
%O A019943 0,2
%A A019943 _N. J. A. Sloane_