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A035610
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G.f.: 3/(1 + 2*sqrt(1-12*x)).
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7
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1, 4, 28, 232, 2092, 19864, 195352, 1970896, 20275660, 211823800, 2240795848, 23951289520, 258255469816, 2805534253552, 30675477376432, 337306474674592, 3727578443380492, 41376874025687032, 461121792658583272, 5157384457905440752, 57869888433073055272, 651266142688270063312, 7349148747954997832272
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OFFSET
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0,2
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COMMENTS
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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, 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, Aug 05 2004
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.
The sequence {b(n)}, where b((2n)=a(n), b(2n+1)=0, is the cogrowth function of the free group of rank 2. - Murray Elder, Jun 28 2016
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LINKS
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FORMULA
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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)*Integral_{x=0..12} x^n*sqrt(x*(12-x))/(16-x). (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, ...
...
D-finite with recurrence: n*a(n) + 2*(9-14*n)*a(n-1) + 96*(2*n-3)*a(n-2) = 0. - R. J. Mathar, Nov 14 2011
P-recurrence confirmed using differential equation (-96*x+10)*g(x) + (-192*x^2+28*x-1)*g'(x) - 6 = 0 satisfied by the generating function. - Robert Israel, Jul 06 2015
a(n) are special values of the hypergeometric function 2F1, in Maple notation: a(n) = 3*12^n*GAMMA(n+1/2)*hypergeom([1,n+1/2],[n+2],3/4)/(4*sqrt(Pi)*(n+1)!), n=0,1,... . - Karol A. Penson, Jul 06 2015
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EXAMPLE
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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
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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);
# Alternative:
f:= gfun:-rectoproc({(-192*n-288)*a(n+1)+(28*n+66)*a(n+2)+(-n-3)*a(n+3)=0, a(0)=1, a(1)=4, a(2)=28}, a(n), remember):
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MATHEMATICA
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CoefficientList[ Series[3/(1 + 2Sqrt[1 - 12x]), {x, 0, 19}], x] (* Robert G. Wilson v, Nov 12 2003 *)
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PROG
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(PARI) x='x+O('x^66); Vec(3/(1+2*sqrt(1-12*x))) \\ Joerg Arndt, Sep 06 2014
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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