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A035610
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G.f.: 3/(1+2*sqrt(1-12*x)).
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5
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1, 4, 28, 232, 2092, 19864, 195352, 1970896, 20275660, 211823800, 2240795848, 23951289520, 258255469816, 2805534253552, 30675477376432, 337306474674592, 3727578443380492, 41376874025687032, 461121792658583272
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Number of walks of length 2n on the 4-regular tree beginning and ending at some fixed vertex. The generating function for the corresponding sequence for the m-regular tree is 2*(m-1)/(m-2+m*sqrt(1-4*(m-1)*x)). When m=2 this reduces to the usual generating function for the central binomial coefficients. - Paul Boddington (psb(AT)maths.warwick.ac.uk), Nov 11 2003
Main diagonal of the array A(0,j)=A(i,0)=1 for i,j>=0 and for i,j>=1 A(i,j)=min{A(i,j-1)+3*A(i-1,j); 3*A(i,j-1)+A(i-1,j)} - Benoit Cloitre (benoit7848c(AT)orange.fr), Aug 05 2004
Hankel transform is A133461. - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Dec 01 2007
Also the number of length 2n words over an alphabet of size 4 that can be built by repeatedly inserting doublets into the initially empty word.
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FORMULA
| a(n) = Sum{k, 0<=k<=n} A039599(n,k)*3^(n-k). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Aug 25 2007
Contribution from Paul Barry (pbarry(AT)wit.ie), Sep 15 2009: (Start)
G.f.: 1/(1-4x*c(3x)), c(x) the g.f. of A000108;
G.f.: 1/(1-4x/(1-3x/(1-3x/(1-3x/(1-3x/(1-.... (continued fraction);
G.f.: 1/(1-4x-12x^2/(1-6x-9x^2/(1-6x-9x^2/(1-6x-9x^2/(1-... (continued fraction).
Integral representation: a(n)=(2/pi)*Int(x^n*sqrt(x(12-x))/(16-x),x,0,12). (End)
a(0) = 1; a(n) = 4/n * Sum_{j=0..n-1} C(2*n,j) * (n-j) * 3^j for n>0.
a(n) = upper left term in M^n, M = an infinite square production matrix as follows:
4, 4, 0, 0, 0, 0,...
3, 3, 3, 0, 0, 0,...
3, 3, 3, 3, 0, 0,...
3, 3, 3, 3, 3, 0,...
3, 3, 3, 3, 3, 3,...
...
- Gary W. Adamson, Jul 15 2011
Conjecture: n*a(n) +2*(9-14*n)*a(n-1) +96*(2*n-3)*a(n-2)=0. - R. J. Mathar, Nov 14 2011
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EXAMPLE
| a(2)=28 because there are 4*4=16 walks whose second step is to return to the starting vertex and 4*3=12 walks whose second step is to move away from the starting vertex.
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MAPLE
| a:= n-> `if` (n=0, 1, 4/n*add (binomial (2*n, j) *(n-j)*3^j, j=0..n-1)):
seq (a(n), n=0..20);
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MATHEMATICA
| CoefficientList[ Series[3/(1 + 2Sqrt[1 - 12x]), {x, 0, 19}], x] (from Robert G. Wilson v Nov 12 2003)
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CROSSREFS
| Cf. A089022. 4th column of A183135.
Sequence in context: A152599 A199579 A089023 * A202824 A046904 A030444
Adjacent sequences: A035607 A035608 A035609 * A035611 A035612 A035613
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KEYWORD
| nonn
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| Edited by Alois P. Heinz (heinz(AT)hs-heilbronn.de), Jan 20 2011
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