cp's OEIS Frontend

A016315 Expansion of g.f. 1/((1 - 2*x)*(1 - 7*x)*(1 - 12*x)).

%I A016315 #42 May 04 2025 15:11:58
%S A016315 1,21,319,4305,55015,683697,8369047,101581473,1227048295,14781074385,
%T A016315 177768357559,2135988547329,25651240368391,307950529031985,
%U A016315 3696355860679255,44362916914251873,532401529073793703,6389144031605054097,76672008158297618935,920080056352830739905
%N A016315 Expansion of g.f. 1/((1 - 2*x)*(1 - 7*x)*(1 - 12*x)).
%H A016315 Vincenzo Librandi, <a href="/A016315/b016315.txt">Table of n, a(n) for n = 0..200</a>
%H A016315 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (21,-122,168).
%F A016315 a(n) = 2*2^n/25 - 49*7^n/25 + 72*12^n/25. - _R. J. Mathar_, Jun 23 2013
%F A016315 From _Vincenzo Librandi_, Jun 26 2013: (Start)
%F A016315 a(n) = 21*a(n-1) - 122*a(n-2) + 168*a(n-3).
%F A016315 a(n) = 19*a(n-1) - 84*a(n-2) + 2^n. (End)
%F A016315 From _Seiichi Manyama_, May 04 2025: (Start)
%F A016315 a(n) = Sum_{k=0..n} 5^k * 2^(n-k) * binomial(n+2,k+2) * Stirling2(k+2,2).
%F A016315 a(n) = Sum_{k=0..n} (-5)^k * 12^(n-k) * binomial(n+2,k+2) * Stirling2(k+2,2). (End)
%F A016315 E.g.f.: exp(2*x)*(2 - 49*exp(5*x) + 72*exp(10*x))/25. - _Stefano Spezia_, May 04 2025
%t A016315 CoefficientList[Series[1 / ((1 - 2 x) (1 - 7 x) (1 - 12 x)), {x, 0, 20}], x] (* _Vincenzo Librandi_, Jun 26 2013 *)
%o A016315 (Magma) I:=[1, 21, 319]; [n le 3 select I[n] else 21*Self(n-1)-122*Self(n-2)+168*Self(n-3): n in [1..20]]; // _Vincenzo Librandi_, Jun 26 2013
%o A016315 (Magma) m:=20; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((1/((1-2*x)*(1-7*x)*(1-12*x))))); // _Vincenzo Librandi_, Jun 26 2013
%o A016315 (PARI) x='x+O('x^99); Vec(1/((1-2*x)*(1-7*x)*(1-12*x))) \\ _Altug Alkan_, Sep 21 2018
%Y A016315 Cf. A016307.
%K A016315 nonn,easy
%O A016315 0,2
%A A016315 _N. J. A. Sloane_