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A001701 Generalized Stirling numbers.

%I A001701 M4169 N1735 #95 Feb 29 2024 15:41:09
%S A001701 1,6,26,71,155,295,511,826,1266,1860,2640,3641,4901,6461,8365,10660,
%T A001701 13396,16626,20406,24795,29855,35651,42251,49726,58150,67600,78156,
%U A001701 89901,102921,117305,133145,150536,169576,190366,213010,237615,264291,293151,324311
%N A001701 Generalized Stirling numbers.
%C A001701 For n>3, a(n-2) gives the number of bounded regions created when the pairwise perpendicular bisectors of n points divide the Euclidean plane into a maximum of A308305(n) regions. This is also equivalent to the number of regions lost from A308305(n) when n>3 points move from maximal position to a circle. - _Alvaro Carbonero_, _Elizabeth Castellano_, _Charles Kulick_, _Karie Schmitz_, Jul 26 2019
%D A001701 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A001701 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A001701 T. D. Noe, <a href="/A001701/b001701.txt">Table of n, a(n) for n = 1..1000</a>
%H A001701 Alvaro Carbonero, Beth Anne Castellano, Gary Gordon, Charles Kulick, Karie Schmitz, and Brittany Shelton, <a href="https://arxiv.org/abs/2106.14140">Permutations of point sets in R_d</a>, arXiv:2106.14140 [math.CO], 2021.
%H A001701 D. S. Mitrinovic and M. S. Mitrinovic, <a href="http://pefmath2.etf.rs/files/47/77.pdf">Tableaux d'une classe de nombres reliés aux nombres de Stirling</a>, Univ. Beograd. Pubi. Elektrotehn. Fak. Ser. Mat. Fiz. 77 (1962).
%H A001701 Robert E. Moritz, <a href="/A001701/a001701.pdf">On the sum of products of n consecutive integers</a>, Univ. Washington Publications in Math., 1 (No. 3, 1926), 44-49 [Annotated scanned copy]
%H A001701 Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H A001701 Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H A001701 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F A001701 a(n) = n*(n-1)*(3n^2 + 17n + 26)/24, n > 1.
%F A001701 G.f.: z*(-1-z-6*z^2+9*z^3-5*z^4+z^5)/(z-1)^5. - _Simon Plouffe_ in his 1992 dissertation
%F A001701 If we define f(n,i,a) = Sum_{k=0..n-i} binomial(n,k)*Stirling1(n-k,i) * Product_{j=0..k-1} (-a - j), then a(n) = f(n,n-2,2), for n >= 2. - _Milan Janjic_, Dec 20 2008
%F A001701 For n>1, a(n) = A308305(n+2) - (n^2 + 3n + 2). - _Alvaro Carbonero_, _Elizabeth Castellano_, _Charles Kulick_, _Karie Schmitz_, Jul 20 2019
%F A001701 E.g.f.: x + (1/24)*exp(x)*x^2*(72 + 32*x + 3*x^2). - _Stefano Spezia_, Sep 07 2019
%F A001701 a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - _Colin Barker_, Jul 08 2020
%p A001701 A001701 := proc(n)
%p A001701     if n = 1 then
%p A001701         1;
%p A001701     else
%p A001701         n*(n-1)*(3*n^2+17*n+26)/24 ;
%p A001701     end if;
%p A001701 end proc: # _R. J. Mathar_, Sep 23 2016
%t A001701 f[k_] := k + 1; t[n_] := Table[f[k], {k, 1, n}]; a[n_] := SymmetricPolynomial[2, t[n]]; Join[{1}, Table[a[n], {n, 2, 30}]] (* _Clark Kimberling_, Dec 31 2011 *)
%t A001701 Join[{1}, Table[n (n - 1) (3 n^2 + 17 n + 26) / 24, {n, 2, 40}]] (* _Vincenzo Librandi_, Sep 30 2018 *)
%t A001701 CoefficientList[Series[(-1 - x - 6 x^2 + 9 x^3 - 5 x^4 + x^5)/(-1 + x)^5, {x, 0, 30}], x] (* _Stefano Spezia_, Sep 30 2018 *)
%t A001701 Prepend[Table[Coefficient[Product[x+j, {j,2,k}], x, k-3], {k,3,40}],1] (* or *) Prepend[LinearRecurrence[{5, -10, 10, -5, 1}, {6, 26, 71, 155, 295}, 40],1] (*_Robert A. Russell_, Oct 04 2018 *)
%o A001701 (GAP) Concatenation([1],List([2..40],n->n*(n-1)*(3*n^2+17*n+26)/24)); # _Muniru A Asiru_, Sep 29 2018
%o A001701 (Magma) [1] cat [n*(n-1)*(3*n^2 + 17*n + 26)/24: n in [2..40]]; // _Vincenzo Librandi_, Sep 30 2018
%o A001701 (PARI) Vec(x*(-1-x-6*x^2+9*x^3-5*x^4+x^5)/(-1+x)^5+O(x^30)) \\ _Stefano Spezia_, Sep 30 2018
%Y A001701 Equals A059302(n+2) + 1, n>1. Partial sums of A005564.
%Y A001701 For n>1, a(n) = A145324(n+1,3).
%K A001701 nonn,easy
%O A001701 1,2
%A A001701 _N. J. A. Sloane_