A001240 Expansion of 1/((1-2x)(1-3x)(1-6x)).

1, 11, 85, 575, 3661, 22631, 137845, 833375, 5019421, 30174551, 181222405, 1087861775, 6528756781, 39177307271, 235078159765, 1410511939775, 8463200647741, 50779591044791, 304678708005925

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

Comments

Differences of reciprocals of unity.

References

• F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 228.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• T. D. Noe, Table of n, a(n) for n = 1..100
• Aung Phone Maw, Aung Kyaw, Recursive Harmonic Numbers and Binomial Coefficients, arXiv:1711.10716 [math.CO], 2017.
• Mircea Merca, Some experiments with complete and elementary symmetric functions, Periodica Mathematica Hungarica, 69 (2014), 182-189.
• Jerry Metzger and Thomas Richards, A Prisoner Problem Variation, Journal of Integer Sequences, Vol. 18 (2015), Article 15.2.7.
• 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 (11,-36,36).

Crossrefs

Right-hand column 2 in triangle A008969.

a(n) = A112492(n+1, 3).

Cf. A021029 (partial sums).

Programs

Maple

A001240:=-1/((6*z-1)*(3*z-1)*(2*z-1)); # conjectured (correctly) by Simon Plouffe in his 1992 dissertation

Mathematica

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 *)

PARI

a(n)=(6^n-2*3^n+2^n)/2 \\ Charles R Greathouse IV, Feb 19 2017

Formula

a(n) = 11a(n-1) - 36a(n-2) + 36a(n-3). - John W. Layman

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

a(n) = Sum_{0<=i,j,k,<=n, i+j+k=n} 2^i*3^j*6^k. - Hieronymus Fischer, Jun 25 2007

a(n) = 2^n + 3^(n+1)*(2^n-1). - Hieronymus Fischer, Jun 25 2007

a(n) = Sum_{k = 0..n-1} 2^(n-2-k) * (3^n - 3^k). - J. M. Bergot, Feb 05 2018