A001240 Expansion of 1/((1-2x)(1-3x)(1-6x)). (Formerly M4798 N2049)

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

Offset

1,2

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).

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

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 *)

Prog

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

Crossrefs

Right-hand column 2 in triangle A008969.

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

Cf. A021029 (partial sums).

Sequence in context: A244975 A271558 A295168 * A129180 A082365 A344480

Adjacent sequences: A001237 A001238 A001239 * A001241 A001242 A001243

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Status

approved