A058396 Expansion of ((1-x)/(1-2*x))^3.

1, 3, 9, 25, 66, 168, 416, 1008, 2400, 5632, 13056, 29952, 68096, 153600, 344064, 765952, 1695744, 3735552, 8192000, 17891328, 38928384, 84410368, 182452224, 393216000, 845152256, 1811939328, 3875536896, 8271167488, 17616076800

Offset: 0

Author

Henry Bottomley, Nov 24 2000

Keywords

• nonn
• easy

Comments

If X_1,X_2,...,X_n are 2-blocks of a (2n+3)-set X then, for n>=1, a(n+1) is the number of (n+2)-subsets of X intersecting each X_i, (i=1,2,...,n). - Milan Janjic, Nov 18 2007

Equals row sums of triangle A152230. - Gary W. Adamson, Nov 29 2008

a(n) is the number of weak compositions of n with exactly 2 parts equal to 0. - Milan Janjic, Jun 27 2010

Except for an initial 1, this is the p-INVERT of (1,1,1,1,1,...) for p(S) = (1 - S)^3; see A291000. - Clark Kimberling, Aug 24 2017

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000
• Robert Davis and Greg Simay, Further Combinatorics and Applications of Two-Toned Tilings, arXiv:2001.11089 [math.CO], 2020.
• Nickolas Hein and Jia Huang, Variations of the Catalan numbers from some nonassociative binary operations, arXiv:1807.04623 [math.CO], 2018.
• Milan Janjic, Two Enumerative Functions.
• Milan Janjić, On Restricted Ternary Words and Insets, arXiv:1905.04465 [math.CO], 2019.
• Milan Janjic and Boris Petkovic, A Counting Function, arXiv preprint arXiv:1301.4550 [math.CO], 2013. - From _N. J. A. Sloane_, Feb 13 2013
• Milan Janjic and Boris Petkovic, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq. 17 (2014), Article 14.3.5.
• Index entries for linear recurrences with constant coefficients, signature (6,-12,8).

Crossrefs

Cf. A045623, A001793, A152230. A diagonal of A058395.

Programs

Magma

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(((1-x)/(1-2*x))^3)); // G. C. Greubel, Oct 16 2018

Maple

seq(coeff(series(((1-x)/(1-2*x))^3,x,n+1), x, n), n = 0 .. 30); # Muniru A Asiru, Oct 16 2018

Mathematica

CoefficientList[ Series[(1 - x)^3/(1 - 2x)^3, {x, 0, 28}], x] (* Robert G. Wilson v, Jun 28 2005 *)
Join[{1},LinearRecurrence[{6,-12,8},{3,9,25},40]] (* Harvey P. Dale, Oct 17 2011 *)

PARI

Vec((1-x)^3/(1-2*x)^3+O(x^99)) \\ Charles R Greathouse IV, Sep 23 2012

Formula

a(n) = (n+2)*(n+7)*2^(n-4) for n > 0.

a(n) = Sum_{k=0..floor((n+2)/2)} C(n+2, 2k)*k(k+1)/2. - Paul Barry, May 15 2003

Binomial transform of quarter squares A002620 (without leading zeros). - Paul Barry, May 27 2003

a(n) = Sum_{k=0..n} C(n, k)*floor((k+2)^2/4). - Paul Barry, May 27 2003

a(n) = 6*a(n-1) - 12*a(n-2) + 8*a(n-3), n > 3. - Harvey P. Dale, Oct 17 2011

From Amiram Eldar, Jan 05 2022: (Start)

Sum_{n>=0} 1/a(n) = 145189/525 - 1984*log(2)/5.

Sum_{n>=0} (-1)^n/a(n) = 30103/175 - 2112*log(3/2)/5. (End)

E.g.f.: (1 + exp(2*x)*(7 + 10*x + 2*x^2))/8. - Stefano Spezia, Feb 01 2025