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%I #30 Sep 08 2022 08:45:08
%S 1,6,46,212,738,2104,5150,11196,22162,40688,70254,115300,181346,
%T 275112,404638,579404,810450,1110496,1494062,1977588,2579554,3320600,
%U 4223646,5314012,6619538,8170704,10000750,12145796,14644962,17540488,20877854
%N Solution to the Dancing School Problem with 5 girls and n+5 boys: f(5,n).
%C f(g,h) = per(B), the permanent of the (0,1)-matrix B of size g X (g+h) with b(i,j)=1 if and only if i <= j <= i+h. See A079908 for more information.
%C For fixed g, f(g,n) is polynomial in n for n >= g-2. See reference.
%H Colin Barker, <a href="/A079910/b079910.txt">Table of n, a(n) for n = 0..1000</a>
%H Jaap Spies, <a href="http://www.jaapspies.nl/mathfiles/dancingschool.pdf">Dancing School Problems, Permanent solutions of Problem 29</a>, Nieuw Archief voor Wiskunde 5/7 nr. 4, Dec 2006, pp. 283-285.
%H Jaap Spies, <a href="http://www.jaapspies.nl/oeis/a079910.sage">Sage program for computing A079910</a>.
%H Jaap Spies, <a href="http://www.jaapspies.nl/mathfiles/dancing.sage">Sage program for computing the polynomial a(n)</a>.
%H Jaap Spies, <a href="http://www.jaapspies.nl/bookb5.pdf">A Bit of Math, The Art of Problem Solving</a>, Jaap Spies Publishers (2019).
%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(0)=1, a(1)=6, a(2)=46, a(n) = n^5 - 5*n^4 + 25*n^3 - 55*n^2 + 80*n - 46.
%F G.f.: (6*x^7 + 11*x^6 + 20*x^5 + 51*x^4 + 6*x^3 + 25*x^2 + 1) / (x-1)^6. - _Colin Barker_, Jan 04 2015
%F E.g.f.: 47 + 6*x + exp(x)*(-46 + 46*x + 20*x^3 + 5*x^4 + x^5). - _Stefano Spezia_, Dec 18 2019
%t CoefficientList[Series[(6 x^7 + 11 x^6 + 20 x^5 + 51 x^4 + 6 x^3 + 25 x^2 + 1) / (x - 1)^6, {x, 0, 30}], x] (* _Vincenzo Librandi_, Feb 17 2015 *)
%o (PARI) Vec((6*x^7+11*x^6+20*x^5+51*x^4+6*x^3+25*x^2+1)/(x-1)^6 + O(x^100)) \\ _Colin Barker_, Jan 04 2015
%o (Magma) [1,6] cat [n^5-5*n^4+25*n^3-55*n^2+80*n-46: n in [2..30]]; // _Vincenzo Librandi_, Feb 17 2015
%Y Cf. A079908-A079928.
%K nonn,easy
%O 0,2
%A _Jaap Spies_, Jan 28 2003
%E More terms from _Benoit Cloitre_, Jan 29 2003
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