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A107264
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Expansion of (1-3x-sqrt((1-3x)^2-4*3*x^2))/(2*3*x^2).
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5
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1, 3, 12, 54, 261, 1323, 6939, 37341, 205011, 1143801, 6466230, 36960300, 213243435, 1240219269, 7263473148, 42799541886, 253556163243, 1509356586897, 9023497273548, 54154973176074, 326154592965879, 1970575690572297
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Series reversion of x/(1+3x+3x^2). Transform of 3^n under the matrix A107131. A row of A107267.
Counts colored Motzkin paths, where H(1,0) and U(1,1) each have 3 colors and D(1,-1) one color. - Paul Barry (pbarry(AT)wit.ie), May 18 2005
Number of Motzkin paths of length n in which both the "up" and the "level" steps come in three colors. - Paul Barry (pbarry(AT)wit.ie), May 18 2005
Third binomial transform of 1,0,3,0,18,0... or 3^n*C(n) (A005159) with interpolated zeros. - Paul Barry (pbarry(AT)wit.ie), May 24 2005
As a continued fraction, the g.f. is 1/(1-3*x-3*x^2/(1-3*x-3*x^2/(1-3*x-3*x^2/(1-3*x-3*x^2/(..... [From Paul Barry (pbarry(AT)wit.ie), Dec 02 2008]
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REFERENCES
| Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011; http://repository.wit.ie/1693/1/AoifeThesis.pdf
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FORMULA
| G.f.: (1-3x-sqrt(1-6x-3x^2))/(6x^2); a(n)=sum{k=0..n, (1/(k+1))C(k+1, n-k+1)C(n, k)3^k}.
a(n)=sum{k=0..floor(n/2), C(n, 2k)C(k)*3^(n-k)} - Paul Barry (pbarry(AT)wit.ie), May 18 2005
E.g.f.: exp(3x)Bessel_I(1, sqrt(3)*2*x)/(sqrt(3)x); - Paul Barry (pbarry(AT)wit.ie), May 24 2005
a(n)=(1/pi)*int(x^n*sqrt(-x^2+6x+3)/6,x,3-2sqrt(3),3+2sqrt(3)); - Paul Barry (pbarry(AT)wit.ie), Sep 16 2006
a(n)=A156016(n+1)/3. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Feb 04 2009]
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CROSSREFS
| Sequence in context: A054666 A006026 A158826 * A200740 A177133 A186241
Adjacent sequences: A107261 A107262 A107263 * A107265 A107266 A107267
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KEYWORD
| easy,nonn,changed
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AUTHOR
| Paul Barry (pbarry(AT)wit.ie), May 15 2005
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