A212337 - OEIS

%I #41 Feb 21 2025 12:31:32

%S 1,8,42,184,731,2736,9844,34448,118101,398584,1328606,4384392,

%T 14348911,46633952,150663528,484275616,1549681961,4939611240,

%U 15690529810,49686677720,156905298051,494251688848,1553362450652,4871909504304,15251194969981,47659984281176

%N Expansion of 1/(1-4*x+3*x^2)^2.

%C Partial sums of A014915. - _Bruno Berselli_, Oct 26 2012

%C Convolution of A003462(n+1) with itself. - _Philippe Deléham_, Mar 07 2014

%H Bruno Berselli, <a href="/A212337/b212337.txt">Table of n, a(n) for n = 0..1000</a>

%H Dillan Agrawal, Selena Ge, Jate Greene, Tanya Khovanova, Dohun Kim, Rajarshi Mandal, Tanish Parida, Anirudh Pulugurtha, Gordon Redwine, Soham Samanta, and Albert Xu, <a href="https://arxiv.org/abs/2501.06675">Chip-Firing on Infinite k-ary Trees</a>, arXiv:2501.06675 [math.CO], 2025. See p. 14.

%H Jean-Luc Baril, Pamela E. Harris, and José L. Ramírez, <a href="https://arxiv.org/abs/2405.05357">Flattened Catalan Words</a>, arXiv:2405.05357 [math.CO], 2024. See p. 16.

%H Sergey Kitaev, Jeffrey Remmel, and Mark Tiefenbruck, <a href="http://arxiv.org/abs/1201.6243">Marked mesh patterns in 132-avoiding permutations I,</a> arXiv:1201.6243v1 [math.CO], 2012. See (16).

%H Sergey Kitaev, Jeffrey Remmel, and Mark Tiefenbruck, <a href="http://www.emis.de/journals/INTEGERS/papers/p16/p16.Abstract.html">Quadrant Marked Mesh Patterns in 132-Avoiding Permutations II</a>, Electronic Journal of Combinatorial Number Theory, Volume 15 #A16. (<a href="http://arxiv.org/abs/1302.2274">arXiv:1302.2274</a>)

%H Mark Shattuck, <a href="https://arxiv.org/abs/2502.10661">Enumeration of consecutive patterns in flattened Catalan words</a>, arXiv:2502.10661 [math.CO], 2025. See pp. 3, 20.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (8,-22,24,-9).

%F From _Bruno Berselli_, May 11 2012: (Start)

%F G.f.: 1/((1-x)^2*(1-3*x)^2).

%F a(n) = 1+n*(1+9*3^n)/4. (End)

%F E.g.f.: exp(x)*(4 + x + 27*exp(2*x)*x)/4. - _Stefano Spezia_, May 14 2024

%e a(0) = 1*1 = 1;

%e a(1) = 1*4 + 4*1 = 8;

%e a(2) = 1*13 + 4*4 + 13*1 = 42;

%e a(3) = 1*40 + 4*13 + 13*4 + 40*1 = 184;

%e a(4) = 1*121 + 4*40 + 13*13 + 40*4 + 121*1 = 731; etc. - _Philippe Deléham_, Mar 07 2014

%t Table[1 + n ((1 + 9 3^n)/4), {n, 0, 25}] (* _Bruno Berselli_, May 11 2012 *)

%t CoefficientList[Series[1/(1-4x+3x^2)^2,{x,0,30}],x] (* or *) LinearRecurrence[ {8,-22,24,-9},{1,8,42,184},30] (* _Harvey P. Dale_, Jun 14 2020 *)

%o (Magma) m:=26; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/((1-x)^2*(1-3*x)^2))); // _Bruno Berselli_, May 11 2012

%o (PARI) Vec(1/(1-4*x+3*x^2)^2 + O(x^100)) \\ _Altug Alkan_, Nov 01 2015

%Y Cf. A003462, A014915.

%K nonn,easy,changed

%O 0,2

%A _N. J. A. Sloane_, May 09 2012