%I #32 Sep 08 2022 08:44:48
%S 1,14,125,910,5901,35574,204205,1132670,6129101,32566534,170691885,
%T 885423630,4556561101,23305343894,118631189165,601616805790,
%U 3042056477901,15346559343654,77279066272045,388583895311150,1951684190615501,9793511186181814,49108010998116525
%N Expansion of 1/((1-2x)(1-3x)(1-4x)(1-5x)).
%C This gives the fourth column of the Sheffer triangle A143494 (2-restricted Stirling2 numbers). See the e.g.f. given below, and comments on the general case under A193685. - _Wolfdieter Lang_, Oct 08 2011
%H Vincenzo Librandi, <a href="/A025211/b025211.txt">Table of n, a(n) for n = 0..500</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (14,-71,154,-120)
%F If we define f(m,j,x) = Sum_{k=j..m} binomial(m,k)*stirling2(k,j)*x^(m-k) then a(n-3) = (-1)^(n-1)*f(n,3,-5), (n >= 3). - _Milan Janjic_, Apr 26 2009
%F If we define f(m,j,x) = Sum_{k=j..m} binomial(m,k)*stirling2(k,j)*x^(m-k) then a(n-3) = f(n,3,2), (n >= 3). - _Milan Janjic_, Apr 26 2009
%F a(n) = 3^(n+3)/2 - 2*4^(n+2) - 2^(n+2)/3 + 5^(n+3)/6. - _R. J. Mathar_, Mar 22 2011
%F O.g.f.: 1/((1-2*x)*(1-3*x)*(1-4*x)*(1-5*x)).
%F E.g.f.: (d^3/dx^3) (exp(2*x)*((exp(x)-1)^3)/3!). See the Sheffer comment given above. - _Wolfdieter Lang_, Oct 08 2011
%t CoefficientList[Series[1/((1 - 2 x) (1 - 3 x) (1 - 4 x) (1 - 5 x)), {x, 0, 25}], x] (* _Vladimir Joseph Stephan Orlovsky_, Jun 20 2011 *)
%t LinearRecurrence[{14,-71,154,-120},{1,14,125,910},30] (* _Harvey P. Dale_, Feb 05 2020 *)
%o (Magma) [3^(n+3)/2 -2*4^(n+2)-2^(n+2)/3+5^(n+3)/6: n in [0..30]]; // _Vincenzo Librandi_, Jun 21 2011
%o (PARI) a(n)=n-=2;3^n*3/2-2*4^n-2^n/3+5^n*5/6 \\ _Charles R Greathouse IV_, Jun 21 2011
%Y Cf. A000079, A001047, A016269.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_