A079911 Solution to the Dancing School Problem with 6 girls and n+6 boys: f(6,n).

1, 7, 79, 478, 2108, 7364, 21652, 55532, 127604, 268108, 523244, 960212, 1672972, 2788724, 4475108, 6948124, 10480772, 15412412, 22158844, 31223108, 43207004, 58823332, 78908852, 104437964, 136537108, 176499884, 225802892, 286122292

Offset: 0

Author

Jaap Spies, Jan 28 2003

Keywords

• nonn
• easy

Comments

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.

For fixed g, f(g,n) is polynomial in n for n >= g-2. See reference.

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Jaap Spies, Dancing School Problems, Nieuw Archief voor Wiskunde 5/7 nr. 4, Dec 2006, pp. 283-285.
• Jaap Spies, Dancing School Problems, Permanent solutions of Problem 29.
• Jaap Spies, Sage program for computing A079911.
• Jaap Spies, Sage program for computing the polynomial a(n).
• Jaap Spies, A Bit of Math, The Art of Problem Solving, Jaap Spies Publishers (2019).
• Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Crossrefs

Cf. A079908-A079928.

Programs

Maple

seq(n^6-9*n^5+60*n^4-225*n^3+555*n^2-774*n+484,n=4..40);

Mathematica

CoefficientList[Series[-(6 x^10 - 29 x^9 + 120 x^8 - 49 x^7 + 267 x^6 + 105 x^5 + 211 x^4 + 37 x^3 + 51 x^2 + 1)/(x - 1)^7, {x, 0, 28}], x] (* Michael De Vlieger, Dec 23 2019 *)
LinearRecurrence[{7,-21,35,-35,21,-7,1},{1,7,79,478,2108,7364,21652,55532,127604,268108,523244},40] (* Harvey P. Dale, Jul 02 2022 *)

PARI

Vec(-(6*x^10 -29*x^9 +120*x^8 -49*x^7 +267*x^6 +105*x^5 +211*x^4 +37*x^3 +51*x^2 +1) / (x -1)^7 + O(x^100)) \\ Colin Barker, Jan 04 2015

Formula

a(0)=1, a(2)=7, a(3)=79, a(n)=n^6-9*n^5+60*n^4-225*n^3+555*n^3-774*n+484.

G.f.: -(6*x^10 -29*x^9 +120*x^8 -49*x^7 +267*x^6 +105*x^5 +211*x^4 +37*x^3 +51*x^2 +1) / (x -1)^7. - Colin Barker, Jan 04 2015