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A122868
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Expansion of 1/sqrt(1-6x-3x^2).
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0
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1, 3, 15, 81, 459, 2673, 15849, 95175, 576963, 3523257, 21640365, 133549155, 827418645, 5143397535, 32063180535, 200367960201, 1254816463923, 7873205412825, 49482344889261, 311457546052659
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Binomial transform of A084609. Central coefficients of (1+3x+3x^2)^n.
The number of free (3,3)-Motzkin paths of length n, where free (k,t)-Motzkin paths are the free Motzkin paths with level steps of weight k and down steps of weight t. For example a(2)=15 because there are 9, 3, 3 paths consisting of two level steps, UD's and DU's, respectively. - Carol J. Wang (cerlined7(AT)hotmail.com), Nov 27 2007
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LINKS
| W. Y. C. Chen, N. Y. Li, L. W. Shapiro and S. H. F. Yan, Matrix identities on weighted partial Motzkin paths, European J. Combinatorics, 28 (2007), 1196-2007.
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FORMULA
| a(n)=sum{k=0..floor(n/2), C(n,2k)*C(2k,k)*3^(n-k)}.
E.g.f. : exp(3x) Bessel_I(0,2sqrt(3)x).
Conjecture: n*a(n) +3*(1-2*n)*a(n-1) +3*(1-n)*a(n-2)=0. - R. J. Mathar, Nov 14 2011
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PROG
| (Maxima) a(n):=coeff(expand((1+3*x+3*x^2)^n), x, n);
makelist(a(n), n, 0, 12);
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CROSSREFS
| Sequence in context: A198628 A084120 A163470 * A015680 A084208 A059271
Adjacent sequences: A122865 A122866 A122867 * A122869 A122870 A122871
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KEYWORD
| easy,nonn
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AUTHOR
| Paul Barry (pbarry(AT)wit.ie), Sep 16 2006
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