This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A062109 #63 Jan 02 2025 13:19:58
%S A062109 1,4,14,44,129,360,968,2528,6448,16128,39680,96256,230656,546816,
%T A062109 1284096,2990080,6909952,15859712,36175872,82051072,185139200,
%U A062109 415760384,929562624,2069889024,4591714304,10150215680,22364028928,49123688448,107592286208,235015241728,512040632320
%N A062109 Expansion of ((1-x)/(1-2*x))^4.
%C A062109 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 A062109 If the offset here is set to zero, the binomial transform of A006918. - _R. J. Mathar_, Jun 29 2009
%C A062109 a(n) is the number of weak compositions of n with exactly 3 parts equal to 0. - _Milan Janjic_, Jun 27 2010
%C A062109 Binomial transform of A002623. - _Carl Najafi_, Jan 22 2013
%C A062109 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 A062109 Harry J. Smith, <a href="/A062109/b062109.txt">Table of n, a(n) for n = 0..200</a>
%H A062109 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 A062109 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 A062109 Milan Janjic, <a href="https://old.pmf.unibl.org/wp-content/uploads/2017/10/enumfor.pdf">Two Enumerative Functions</a>
%H A062109 Milan 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 A062109 Milan Janjic and 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 A062109 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (8,-24,32,-16).
%F A062109 a(n) = (n+5)*(n^2 + 13*n + 18)*2^(n-5)/3, with a(0)=1.
%F A062109 a(n) = A055809(n-5)*2^(n-4).
%F A062109 a(n) = 2*a(n-1) + A058396(n) - A058396(n-1).
%F A062109 a(n) = Sum_{k<n} a(k) + A058396(n).
%F A062109 a(n) = A062110(4, n).
%F A062109 G.f.: (1-x)^4/(1-2*x)^4.
%p A062109 seq(coeff(series(((1-x)/(1-2*x))^4, x,n+1),x,n),n=0..30); # _Muniru A Asiru_, Jul 01 2018
%t A062109 CoefficientList[Series[(1 - x)^4/(1 - 2 x)^4, {x, 0, 26}], x] (* _Michael De Vlieger_, Jul 01 2018 *)
%t A062109 LinearRecurrence[{8,-24,32,-16},{1,4,14,44,129},30] (* _Harvey P. Dale_, Sep 02 2022 *)
%o A062109 (PARI) a(n)=if(n<1,n==0,(n+5)*(n^2+13*n+18)*2^n/96)
%o A062109 (Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(((1-x)/(1-2*x))^4)); // _G. C. Greubel_, Oct 16 2018
%K A062109 nonn,easy
%O A062109 0,2
%A A062109 _Henry Bottomley_, May 30 2001