A005337 Number of ways in which n identical balls can be distributed among 4 boxes in a row such that each pair of adjacent boxes contains at least 4 balls.
15, 40, 76, 124, 185, 260, 350, 456, 579, 720, 880, 1060, 1261, 1484, 1730, 2000, 2295, 2616, 2964, 3340, 3745, 4180, 4646, 5144, 5675, 6240, 6840, 7476, 8149, 8860, 9610, 10400, 11231, 12104, 13020, 13980, 14985
Offset: 8
References
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Links
- Vincenzo Librandi, Table of n, a(n) for n = 8..1000
- D. R. Breach, Letter to N. J. A. Sloane, Jun 1980
- Philippe Flajolet, Balls and Urns, etc., A problem in submarine detection (solution to problem 68-16).
- M. Hayes (proposer) and D. R. Breach (solver), A combinatorial problem, Problem 68-16, SIAM Rev. 12 (1970), 294-297.
- Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
- Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
- Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).
Programs
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Maple
A005337:=(15-20*z+6*z**2)/(z-1)**4; # Simon Plouffe in his 1992 dissertation
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Mathematica
CoefficientList[Series[(15 - 20 x + 6 x^2)/(1 - x)^4, {x, 0, 40}], x] (* Vincenzo Librandi, Oct 14 2013 *) LinearRecurrence[{4,-6,4,-1},{15,40,76,124},50] (* Harvey P. Dale, May 11 2014 *)
Formula
G.f.: x^8*(15 - 20*x + 6*x^2)/(1 - x)^4.
a(n) = (546 - 169*n + 6*n^2 + n^3)/6. [Colin Barker, Jul 08 2012]
Extensions
G.f. corrected by Colin Barker, Jul 08 2012
Name clarified by Alois P. Heinz, Oct 02 2017