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%I #71 Sep 08 2022 08:45:02
%S 1,3,9,25,66,168,416,1008,2400,5632,13056,29952,68096,153600,344064,
%T 765952,1695744,3735552,8192000,17891328,38928384,84410368,182452224,
%U 393216000,845152256,1811939328,3875536896,8271167488,17616076800
%N Expansion of ((1-x)/(1-2*x))^3.
%C 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
%C Equals row sums of triangle A152230. - _Gary W. Adamson_, Nov 29 2008
%C a(n) is the number of weak compositions of n with exactly 2 parts equal to 0. - _Milan Janjic_, Jun 27 2010
%C 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
%H Harvey P. Dale, <a href="/A058396/b058396.txt">Table of n, a(n) for n = 0..1000</a>
%H Robert Davis and Greg Simay, <a href="https://arxiv.org/abs/2001.11089">Further Combinatorics and Applications of Two-Toned Tilings</a>, arXiv:2001.11089 [math.CO], 2020.
%H Nickolas Hein and Jia Huang, <a href="https://arxiv.org/abs/1807.04623">Variations of the Catalan numbers from some nonassociative binary operations</a>, arXiv:1807.04623 [math.CO], 2018.
%H Milan Janjic, <a href="https://pmf.unibl.org/wp-content/uploads/2017/10/enumfor.pdf">Two Enumerative Functions</a>.
%H Milan Janjić, <a href="https://arxiv.org/abs/1905.04465">On Restricted Ternary Words and Insets</a>, arXiv:1905.04465 [math.CO], 2019.
%H Milan Janjic and Boris Petkovic, <a href="http://arxiv.org/abs/1301.4550">A Counting Function</a>, arXiv preprint arXiv:1301.4550 [math.CO], 2013. - From _N. J. A. Sloane_, Feb 13 2013
%H Milan Janjic and Boris Petkovic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Janjic/janjic45.html">A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers</a>, J. Int. Seq. 17 (2014), Article 14.3.5.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (6,-12,8).
%F a(n) = (n+2)*(n+7)*2^(n-4) for n > 0.
%F a(n) = Sum_{k=0..floor((n+2)/2)} C(n+2, 2k)*k(k+1)/2. - _Paul Barry_, May 15 2003
%F Binomial transform of quarter squares A002620 (without leading zeros). - _Paul Barry_, May 27 2003
%F a(n) = Sum_{k=0..n} C(n, k)*floor((k+2)^2/4). - _Paul Barry_, May 27 2003
%F a(n) = 6*a(n-1) - 12*a(n-2) + 8*a(n-3), n > 3. - _Harvey P. Dale_, Oct 17 2011
%F From _Amiram Eldar_, Jan 05 2022: (Start)
%F Sum_{n>=0} 1/a(n) = 145189/525 - 1984*log(2)/5.
%F Sum_{n>=0} (-1)^n/a(n) = 30103/175 - 2112*log(3/2)/5. (End)
%p seq(coeff(series(((1-x)/(1-2*x))^3,x,n+1), x, n), n = 0 .. 30); # _Muniru A Asiru_, Oct 16 2018
%t CoefficientList[ Series[(1 - x)^3/(1 - 2x)^3, {x, 0, 28}], x] (* _Robert G. Wilson v_, Jun 28 2005 *)
%t Join[{1},LinearRecurrence[{6,-12,8},{3,9,25},40]] (* _Harvey P. Dale_, Oct 17 2011 *)
%o (PARI) Vec((1-x)^3/(1-2*x)^3+O(x^99)) \\ _Charles R Greathouse IV_, Sep 23 2012
%o (Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(((1-x)/(1-2*x))^3)); // _G. C. Greubel_, Oct 16 2018
%Y Cf. A045623, A001793, A152230. A diagonal of A058395.
%K nonn,easy
%O 0,2
%A _Henry Bottomley_, Nov 24 2000