cp's OEIS Frontend

A019757 Expansion of 1/((1-5*x)(1-6*x)(1-7*x)).

%I A019757 #28 Mar 13 2024 20:11:35
%S A019757 1,18,217,2190,19981,170898,1398097,11075670,85654261,650280378,
%T A019757 4865931577,35994162750,263799130141,1918854559458,13871649322657,
%U A019757 99770067275430,714554190919621,5099624595840138,36287658424563337,257574399980365710,1824480913344734701,12900603911464394418
%N A019757 Expansion of 1/((1-5*x)(1-6*x)(1-7*x)).
%C A019757 Column sequence for m=1 (without leading zeros) of the Sheffer triangle (exp(5*x), exp(x)-1) of 5-restricted Stirling2 numbers. See A193685. - _Wolfdieter Lang_, Oct 07 2011
%H A019757 Vincenzo Librandi, <a href="/A019757/b019757.txt">Table of n, a(n) for n = 0..1000</a>
%H A019757 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (18,-107,210).
%F A019757 If we define f(m,j,x) = Sum_{k=j..m} binomial(m,k)*Stirling2(k,j)*x^(m-k) then a(n-2) = f(n,2,5), (n >= 2). - _Milan Janjic_, Apr 26 2009
%F A019757 E.g.f.: (d^2/dx^2)(exp(5*x)*(exp(x)-1)^2/2!). See the Sheffer triangle comment above. - _Wolfdieter Lang_, Oct 07 2011
%F A019757 From _Vincenzo Librandi_, Oct 08 2011: (Start)
%F A019757 a(n) = (7^(n+2) + 5^(n+2) - 2*6^(n+2))/2.
%F A019757 a(n) = 18*a(n-1) - 107*a(n-2) + 210*a(n-3), n >= 3.
%F A019757 a(n) = 13*a(n-1) - 42*a(n-2) + 5^n, a(0)=1, a(1)=18. (End)
%F A019757 E.g.f.: (49*exp(7*x) + 25*exp(5*x) - 72*exp(6*x))/2. - _G. C. Greubel_, Feb 07 2018
%t A019757 Table[(7^(n+2) + 5^(n+2) - 2*6^(n+2))/2, {n,0,30}] (* _G. C. Greubel_, Feb 07 2018 *)
%t A019757 LinearRecurrence[{18,-107,210},{1,18,217},30] (* _Harvey P. Dale_, Mar 13 2024 *)
%o A019757 (Magma) [(7^(n+2)+5^(n+2)-2*6^(n+2))/2: n in [0..20]]; // _Vincenzo Librandi_, Oct 08 2011
%o A019757 (PARI) for(n=0,30, print1((7^(n+2) + 5^(n+2) - 2*6^(n+2))/2, ", ")) \\ _G. C. Greubel_, Feb 07 2018
%K A019757 nonn,easy
%O A019757 0,2
%A A019757 _N. J. A. Sloane_