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 A001701 Generalized Stirling numbers. (Formerly M4169 N1735) 19
 1, 6, 26, 71, 155, 295, 511, 826, 1266, 1860, 2640, 3641, 4901, 6461, 8365, 10660, 13396, 16626, 20406, 24795, 29855, 35651, 42251, 49726, 58150, 67600, 78156, 89901, 102921, 117305, 133145, 150536, 169576, 190366, 213010, 237615, 264291, 293151, 324311 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS 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 REFERENCES N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS T. D. Noe, Table of n, a(n) for n = 1..1000 Alvaro Carbonero, Beth Anne Castellano, Gary Gordon, Charles Kulick, Karie Schmitz, and Brittany Shelton, Permutations of point sets in R_d, arXiv:2106.14140 [math.CO], 2021. D. S. Mitrinovic and M. S. Mitrinovic, Tableaux d'une classe de nombres reliés aux nombres de Stirling, Univ. Beograd. Pubi. Elektrotehn. Fak. Ser. Mat. Fiz. 77 (1962). Robert E. Moritz, On the sum of products of n consecutive integers, Univ. Washington Publications in Math., 1 (No. 3, 1926), 44-49 [Annotated scanned copy] Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009. Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992 Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1). FORMULA a(n) = n*(n-1)*(3n^2 + 17n + 26)/24, n > 1. 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 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 For n>1, a(n) = A308305(n+2) - (n^2 + 3n + 2). - Alvaro Carbonero, Elizabeth Castellano, Charles Kulick, Karie Schmitz, Jul 20 2019 E.g.f.: x + (1/24)*exp(x)*x^2*(72 + 32*x + 3*x^2). - Stefano Spezia, Sep 07 2019 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 MAPLE A001701 := proc(n)     if n = 1 then         1;     else         n*(n-1)*(3*n^2+17*n+26)/24 ;     end if; end proc: # R. J. Mathar, Sep 23 2016 MATHEMATICA 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 *) Join[{1}, Table[n (n - 1) (3 n^2 + 17 n + 26) / 24, {n, 2, 40}]] (* Vincenzo Librandi, Sep 30 2018 *) 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 *) From Robert A. Russell, Oct 04 2018: (Start) Prepend[Table[Coefficient[Product[x+j, {j, 2, k}], x, k-3], {k, 3, 40}], 1] Prepend[LinearRecurrence[{5, -10, 10, -5, 1}, {6, 26, 71, 155, 295}, 40], 1] (End) PROG (GAP) Concatenation([1], List([2..40], n->n*(n-1)*(3*n^2+17*n+26)/24)); # Muniru A Asiru, Sep 29 2018 (MAGMA) [1] cat [n*(n-1)*(3*n^2 + 17*n + 26)/24: n in [2..40]]; // Vincenzo Librandi, Sep 30 2018 (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 CROSSREFS Equals A059302(n+2) + 1, n>1. Partial sums of A005564. For n>1, a(n) = A145324(n+1,3). Sequence in context: A190095 A135036 A166796 * A241452 A175898 A255870 Adjacent sequences:  A001698 A001699 A001700 * A001702 A001703 A001704 KEYWORD nonn,easy AUTHOR STATUS approved

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Last modified July 1 08:17 EDT 2022. Contains 354953 sequences. (Running on oeis4.)