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%I M4798 N2049 #61 Apr 13 2022 13:25:15
%S 1,11,85,575,3661,22631,137845,833375,5019421,30174551,181222405,
%T 1087861775,6528756781,39177307271,235078159765,1410511939775,
%U 8463200647741,50779591044791,304678708005925
%N Expansion of 1/((1-2x)(1-3x)(1-6x)).
%C Differences of reciprocals of unity.
%D F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 228.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A001240/b001240.txt">Table of n, a(n) for n = 1..100</a>
%H Aung Phone Maw, Aung Kyaw, <a href="https://arxiv.org/abs/1711.10716">Recursive Harmonic Numbers and Binomial Coefficients</a>, arXiv:1711.10716 [math.CO], 2017.
%H Mircea Merca, <a href="http://dx.doi.org/10.1007/s10998-014-0034-3">Some experiments with complete and elementary symmetric functions</a>, Periodica Mathematica Hungarica, 69 (2014), 182-189.
%H Jerry Metzger and Thomas Richards, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Metzger/metz1.html">A Prisoner Problem Variation</a>, Journal of Integer Sequences, Vol. 18 (2015), Article 15.2.7.
%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (11,-36,36).
%F a(n) = 11a(n-1) - 36a(n-2) + 36a(n-3). - _John W. Layman_
%F a(n) = (6^n - 2*3^n + 2^n)/2. Also -x^2/6*Beta(x, 4) = Sum_{n>=0} a(n)*(-x/6)^n. Thus x^2*Beta(x, 4) = x - 11/6*x^2 + 85/36*x^3 - 575/216*x^4 + 3661/1296*x^5 - ... . - _Vladeta Jovovic_, Aug 09 2002
%F a(n) = Sum_{0<=i,j,k,<=n, i+j+k=n} 2^i*3^j*6^k. - _Hieronymus Fischer_, Jun 25 2007
%F a(n) = 2^n + 3^(n+1)*(2^n-1). - _Hieronymus Fischer_, Jun 25 2007
%F a(n) = Sum_{k = 0..n-1} 2^(n-2-k) * (3^n - 3^k). - _J. M. Bergot_, Feb 05 2018
%p A001240:=-1/((6*z-1)*(3*z-1)*(2*z-1)); # conjectured (correctly) by _Simon Plouffe_ in his 1992 dissertation
%t CoefficientList[Series[1/((1-2x)(1-3x)(1-6x)),{x,0,25}],x] (* or *) LinearRecurrence[{11,-36,36},{1,11,85},25] (* _Harvey P. Dale_, May 15 2011 *)
%o (PARI) a(n)=(6^n-2*3^n+2^n)/2 \\ _Charles R Greathouse IV_, Feb 19 2017
%Y Right-hand column 2 in triangle A008969.
%Y a(n) = A112492(n+1, 3).
%Y Cf. A021029 (partial sums).
%K nonn,easy,nice
%O 1,2
%A _N. J. A. Sloane_
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