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A079309
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a(n) = C(1,1)+C(3,2)+C(5,3)+...+C(2n-1,n).
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7
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1, 4, 14, 49, 175, 637, 2353, 8788, 33098, 125476, 478192, 1830270, 7030570, 27088870, 104647630, 405187825, 1571990935, 6109558585, 23782190485, 92705454895, 361834392115, 1413883873975, 5530599237775, 21654401079325
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| a(n) is the sum of pyramid weights of all Dyck paths of length 2n (for pyramid weight see Denise and Simion). Equivalently, a(n) is the sum of the total lengths of end branches of an ordered tree, summation being over all ordered trees with n edges. For example, the five ordered trees with 3 edges have total lengths of endbranches 3,2,3,3 and 3. - Emeric Deutsch (deutsch(AT)duke.poly.edu), May 30 2003
a(n) is the number of Motzkin paths of length 2n with exactly one level segment. (A level segment is a maximal sequence of contiguous flatsteps.) Example: for n=2, the paths counted are FFFF, FFUD, UDFF, UFFD. The formula for a(n) below counts these paths by length of the level segment. - David Callan (callan(AT)stat.wisc.edu), Jul 15 2004
The inverse Catalan transform yields A024495, shifted once left. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 07 2009]
Contribution from Paul Barry (pbarry(AT)wit.ie), Mar 29 2010: (Start)
Hankel transform is A138341.
The aerated sequence 0,0,1,0,4,0,14,0,49,... has e.g.f. int(cosh(x-t)*Bessel_I(1,2t),t,0,x). (End)
a(n) is the number of terms of A031443 not exceeding 4^n. [From Vladimir Shevelev (shevelev(AT)bgu.ac.il), Oct 01 2010]
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REFERENCES
| A. Denise and R. Simion, Two combinatorial statistics on Dyck paths, Discrete Math., 137, 1995, 155-176.
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LINKS
| Guo-Niu Han, Enumeration of Standard Puzzles
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FORMULA
| a(n) = (1/2)*(C(2, 1) + C(4, 2) + C(6, 3) + ... + C(2n, n)) = A066796(n)/2. - Vladeta Jovovic (vladeta(AT)eunet.rs), Feb 12 2003
G.f.: (1/sqrt(1-4*x)-1)/(1-x)/2. - Vladeta Jovovic (vladeta(AT)eunet.rs), Feb 12 2003
Given g.f. A(x), then x*A(x-x^2) is g.f. of A024495. - Michael Somos Feb 14 2006
sum(igcd(binomial(2*j,j))/2,j=1..n). - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Oct 25 2006
a(n)=sum_{0<=i<=<j<=n}binomial(i+j,i) - Benoit Cloitre (benoit7848c(AT)orange.fr), Nov 25 2006
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EXAMPLE
| a(4) = C(1,1)+C(3,2)+C(5,3)+C(7,4) = 1+3+10+35 = 49
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MAPLE
| a:=n->sum(igcd(binomial(2*j, j))/2, j=1..n): seq(a(n), n=1..24); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Oct 25 2006
a:=n->sum(abs(binomial(-j, -2*j)), j=1..n): seq(a(n), n=1..24); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Oct 03 2007
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PROG
| (PARI) a(n)=sum(k=1, n, binomial(2*k-1, k)) /* Michael Somos Feb 14 2006 */
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CROSSREFS
| Cf. A066796, A001700.
Equals A024718(n) - 1.
Sequence in context: A071737 A071741 A001894 * A026630 A034459 A120747
Adjacent sequences: A079306 A079307 A079308 * A079310 A079311 A079312
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KEYWORD
| easy,nonn
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AUTHOR
| Miklos Kristof (kristofmiki(AT)freemail.hu), Feb 10 2003
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EXTENSIONS
| More terms from Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Feb 11 2003
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