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%I #27 Dec 03 2021 15:44:11
%S 1,8,133,1044,5794,24720,86608,260720,693552,1666000,3675680,7549488,
%T 14591440,26770832,46955760,79197040,129067568,204062160,314062912,
%U 471875120,693838800,1000520848,1417492880,1976199792,2714924080
%N Solution to the Dancing School Problem with 7 girls and n+7 boys: f(7,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="/A079912/b079912.txt">Table of n, a(n) for n = 0..1000</a>
%H Jaap Spies, <a href="http://www.nieuwarchief.nl/serie5/pdf/naw5-2006-07-4-283.pdf">Dancing School Problems</a>, Nieuw Archief voor Wiskunde 5/7 nr. 4, Dec 2006, pp. 283-285.
%H Jaap Spies, <a href="http://www.jaapspies.nl/mathfiles/dancingschool.pdf">Dancing School Problems, Permanent solutions of Problem 29</a>.
%H Jaap Spies, <a href="http://www.jaapspies.nl/oeis/a079912.sage">Sage program for computing A079912</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_08">Index entries for linear recurrences with constant coefficients</a>, signature (8,-28,56,-70,56,-28,8,-1).
%F a(0) = 1, a(1) = 8, a(2) = 133, a(3) = 1044, a(4) = 5794; for n>4, a(n) = n^7-14*n^6+126*n^5-700*n^4+2625*n^3-6342*n^2+9072*n-5840.
%F G.f.: -(46*x^12 -340*x^11 +931*x^10 -1808*x^9 +727*x^8 -1400*x^7 -1506*x^6 -656*x^5 -788*x^4 -148*x^3 -97*x^2 -1) / (x -1)^8. - _Colin Barker_, Jan 04 2015
%p seq(n^7-14*n^6+126*n^5-700*n^4+2625*n^3-6342*n^2+9072*n-5840,n=5..20);
%t Join[{1,8,133,1044,5794},Table[n^7-14n^6+126n^5-700n^4+2625n^3- 6342n^2 +9072n-5840,{n,5,30}]] (* _Harvey P. Dale_, May 03 2011 *)
%o (PARI) Vec(-(46*x^12 -340*x^11 +931*x^10 -1808*x^9 +727*x^8 -1400*x^7 -1506*x^6 -656*x^5 -788*x^4 -148*x^3 -97*x^2 -1) / (x -1)^8 + O(x^100)) \\ _Colin Barker_, Jan 04 2015
%Y Cf. A079908-A079928.
%K nonn,easy
%O 0,2
%A _Jaap Spies_, Jan 28 2003
%E More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 06 2003
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