A212337 Expansion of 1/(1-4*x+3*x^2)^2.

1, 8, 42, 184, 731, 2736, 9844, 34448, 118101, 398584, 1328606, 4384392, 14348911, 46633952, 150663528, 484275616, 1549681961, 4939611240, 15690529810, 49686677720, 156905298051, 494251688848, 1553362450652, 4871909504304, 15251194969981, 47659984281176

Offset: 0

Author

N. J. A. Sloane, May 09 2012

Keywords

• nonn
• easy

Comments

Partial sums of A014915. - Bruno Berselli, Oct 26 2012

Convolution of A003462(n+1) with itself. - Philippe Deléham, Mar 07 2014

Examples

			a(0) = 1*1 = 1;
a(1) = 1*4 + 4*1 = 8;
a(2) = 1*13 + 4*4 + 13*1 = 42;
a(3) = 1*40 + 4*13 + 13*4 + 40*1 = 184;
a(4) = 1*121 + 4*40 + 13*13 + 40*4 + 121*1 = 731; etc. - _Philippe Deléham_, Mar 07 2014
		

Links

• Bruno Berselli, Table of n, a(n) for n = 0..1000
• Dillan Agrawal, Selena Ge, Jate Greene, Tanya Khovanova, Dohun Kim, Rajarshi Mandal, Tanish Parida, Anirudh Pulugurtha, Gordon Redwine, Soham Samanta, and Albert Xu, Chip-Firing on Infinite k-ary Trees, arXiv:2501.06675 [math.CO], 2025. See p. 14.
• Jean-Luc Baril, Pamela E. Harris, and José L. Ramírez, Flattened Catalan Words, arXiv:2405.05357 [math.CO], 2024. See p. 16.
• Sergey Kitaev, Jeffrey Remmel, and Mark Tiefenbruck, Marked mesh patterns in 132-avoiding permutations I, arXiv:1201.6243v1 [math.CO], 2012. See (16).
• Sergey Kitaev, Jeffrey Remmel, and Mark Tiefenbruck, Quadrant Marked Mesh Patterns in 132-Avoiding Permutations II, Electronic Journal of Combinatorial Number Theory, Volume 15 #A16. (arXiv:1302.2274)
• Mark Shattuck, Enumeration of consecutive patterns in flattened Catalan words, arXiv:2502.10661 [math.CO], 2025. See pp. 3, 20.
• Index entries for linear recurrences with constant coefficients, signature (8,-22,24,-9).

Crossrefs

Cf. A003462, A014915.

Programs

Magma

m:=26; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/((1-x)^2*(1-3*x)^2))); // Bruno Berselli, May 11 2012

Mathematica

Table[1 + n ((1 + 9 3^n)/4), {n, 0, 25}] (* Bruno Berselli, May 11 2012 *)
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 *)

PARI

Vec(1/(1-4*x+3*x^2)^2 + O(x^100)) \\ Altug Alkan, Nov 01 2015

Formula

From Bruno Berselli, May 11 2012: (Start)

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

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

E.g.f.: exp(x)*(4 + x + 27*exp(2*x)*x)/4. - Stefano Spezia, May 14 2024