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A082490
Exponent of highest power of 3 dividing sum(0<=k<n, C(2n,n)).
1
0, 1, 2, 0, 2, 3, 1, 2, 4, 0, 1, 2, 0, 3, 4, 2, 3, 5, 1, 2, 3, 1, 3, 4, 2, 3, 6, 0, 1, 2, 0, 2, 3, 1, 2, 4, 0, 1, 2, 0, 4, 5, 3, 4, 6, 2, 3, 4, 2, 4, 5, 3, 4, 7, 1, 2, 3, 1, 3, 4, 2, 3, 5, 1, 2, 3, 1, 4, 5, 3, 4, 6, 2, 3, 4, 2, 4, 5, 3, 4, 8, 0, 1, 2, 0, 2, 3, 1, 2, 4, 0, 1, 2, 0, 3, 4
OFFSET
1,3
LINKS
J.-P. Allouche and J. Shallit, The ring of k-regular sequences, II, Theoret. Computer Sci., 307 (2003), 3-29.
J. Shallit, Number theory and formal languages, in D. A. Hejhal, J. Friedman, M. C. Gutzwiller and A. M. Odlyzko, eds., Emerging Applications of Number Theory, IMA Volumes in Mathematics and Its Applications, V. 109, Springer-Verlag, 1999, pp. 547-570. (Example 1.)
FORMULA
a(n) = A007949(A006134(n)) = A007949 (n^2 * C(2n, n)) (Allouche, Shallit; Zagier) = 2*A007949(n) + A000989(n).
MAPLE
map(t -> padic:-ordp(t, 3), ListTools:-PartialSums([seq(binomial(2*n, n), n=0..100)])); # Robert Israel, Mar 27 2018
PROG
(PARI) s=0; for(n=1, 150, s=s+binomial(2*n-2, n-1); print1(valuation(s, 3)", "))
(PARI) a(n) = valuation(n^2 * binomial(2*n, n), 3); \\ Michel Marcus, Mar 27 2018
CROSSREFS
KEYWORD
nonn,hear
AUTHOR
Ralf Stephan, Apr 28 2003
STATUS
approved