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Expansion of ((1-x)/(1-2*x))^4.
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%I #56 Sep 08 2022 08:45:03

%S 1,4,14,44,129,360,968,2528,6448,16128,39680,96256,230656,546816,

%T 1284096,2990080,6909952,15859712,36175872,82051072,185139200,

%U 415760384,929562624,2069889024,4591714304,10150215680,22364028928

%N Expansion of ((1-x)/(1-2*x))^4.

%C If X_1,X_2,...,X_n are 2-blocks of a (2n+4)-set X then, for n >= 1, a(n+1) is the number of (n+3)-subsets of X intersecting each X_i, (i=1,2,...,n). - _Milan Janjic_, Nov 23 2007

%C If the offset here is set to zero, the binomial transform of A006918. - _R. J. Mathar_, Jun 29 2009

%C a(n) is the number of weak compositions of n with exactly 3 parts equal to 0. - _Milan Janjic_, Jun 27 2010

%C Binomial transform of A002623. - _Carl Najafi_, Jan 22 2013

%C Except for an initial 1, this is the p-INVERT of (1,1,1,1,1,...) for p(S) = (1 - S)^4; see A291000. - _Clark Kimberling_, Aug 24 2017

%H Harry J. Smith, <a href="/A062109/b062109.txt">Table of n, a(n) for n = 0..200</a>

%H Robert Davis, 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, 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="http://www.pmfbl.org/janjic/">Two Enumerative Functions</a>

%H M. Janjic and B. 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 M. Janjic, B. 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) # 14.3.5.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (8,-24,32,-16).

%F a(n) = (n+5)*(n^2 + 13*n + 18)*2^(n-5)/3, with a(0)=1.

%F a(n) = A055809(n-5)*2^(n-4).

%F a(n) = 2*a(n-1) + A058396(n) - A058396(n-1).

%F a(n) = Sum_{k<n} a(k) + A058396(n).

%F a(n) = A062110(4, n).

%F G.f.: (1-x)^4/(1-2*x)^4.

%p seq(coeff(series(((1-x)/(1-2*x))^4, x,n+1),x,n),n=0..30); # _Muniru A Asiru_, Jul 01 2018

%t CoefficientList[Series[(1 - x)^4/(1 - 2 x)^4, {x, 0, 26}], x] (* _Michael De Vlieger_, Jul 01 2018 *)

%t LinearRecurrence[{8,-24,32,-16},{1,4,14,44,129},30] (* _Harvey P. Dale_, Sep 02 2022 *)

%o (PARI) a(n)=if(n<1,n==0,(n+5)*(n^2+13*n+18)*2^n/96)

%o (PARI) { a=1; for (n=0, 200, if (n, a=(n + 5)*(n^2 + 13*n + 18)*2^n/96); write("b062109.txt", n, " ", a) ) } \\ _Harry J. Smith_, Aug 01 2009

%o (Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(((1-x)/(1-2*x))^4)); // _G. C. Greubel_, Oct 16 2018

%K nonn,easy

%O 0,2

%A _Henry Bottomley_, May 30 2001