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%I #42 Dec 28 2023 14:48:55
%S 1,2,4,8,16,32,64,126,240,438,764,1276,2048,3172,4760,6946,9888,13770,
%T 18804,25232,33328,43400,55792,70886,89104,110910,136812,167364,
%U 203168,244876,293192,348874,412736,485650,568548,662424,768336,887408,1020832,1169870
%N Maximal number of regions into which 5-space can be divided by n hyperspheres.
%C n hyperspheres divide R^k into at most binomial(n-1, k) + Sum_{i=0..k} binomial(n, i) regions.
%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 73, Problem 4.
%H Muniru A Asiru, <a href="/A059174/b059174.txt">Table of n, a(n) for n = 0..10000</a>
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1).
%F a(n) = binomial(n-1, 5) + Sum_{i=0..5} binomial(n, i).
%F G.f.: (x^6 + 3*x^4 - 6*x^3 + 7*x^2 - 4*x + 1)/(x - 1)^6. - _Colin Barker_, Oct 06 2012
%F a(n) = 2*A006261(n-1), for n > 0. - _Günter Rote_, Dec 18 2018, by elementary manipulations.
%F E.g.f.: 1 + (1/60)*(120*x + 20*x^3 + x^5)*exp(x). - _Franck Maminirina Ramaharo_, Dec 21 2018
%p seq(coeff(series((x^6+3*x^4-6*x^3+7*x^2-4*x+1)/(1-x)^6,x,n+1), x, n), n = 0 .. 40); # _Muniru A Asiru_, Dec 18 2018
%t Join[{1}, Table[((n^5 - 5 n^4 + 25 n^3 + 5 n^2 + 94 n + 120) / 60), {n, 0, 50}]] (* _Vincenzo Librandi_, Dec 21 2018 *)
%o (PARI) a(n) = binomial(n-1, 5) + sum(i=0, 5, binomial(n, i)); \\ _Michel Marcus_, Jan 29 2016
%o (GAP) Concatenation([1], List([1..40], n-> Binomial(n-1,5) + Sum([0..5], i-> Binomial(n,i)))); # _Muniru A Asiru_, Dec 18 2018
%o (Magma) [1] cat [(n^5-5*n^4+25*n^3+5*n^2+94*n+120)/60: n in [0..40]]; // _Vincenzo Librandi_, Dec 21 2018
%Y Cf. A014206 (dim 2), A046127 (dim 3), A059173 (dim 4), this sequence (dim 5).
%Y Fifth row (k=5) of A059250.
%Y Cf. A006261.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_, Feb 15 2001
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