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A001701
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Generalized Stirling numbers.
(Formerly M4169 N1735)
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19
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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
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OFFSET
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1,2
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COMMENTS
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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
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REFERENCES
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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).
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LINKS
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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.
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FORMULA
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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
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
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MAPLE
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if n = 1 then
1;
else
n*(n-1)*(3*n^2+17*n+26)/24 ;
end if;
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MATHEMATICA
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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 *)
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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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