This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A079309 #127 Jun 19 2025 00:13:48
%S A079309 1,4,14,49,175,637,2353,8788,33098,125476,478192,1830270,7030570,
%T A079309 27088870,104647630,405187825,1571990935,6109558585,23782190485,
%U A079309 92705454895,361834392115,1413883873975,5530599237775,21654401079325,84859704298201,332818970772253
%N A079309 a(n) = C(1,1) + C(3,2) + C(5,3) + ... + C(2*n-1,n).
%C A079309 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_, May 30 2003
%C A079309 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_, Jul 15 2004
%C A079309 The inverse Catalan transform yields A024495, shifted once left. - _R. J. Mathar_, Jul 07 2009
%C A079309 From _Paul Barry_, Mar 29 2010: (Start)
%C A079309 Hankel transform is A138341.
%C A079309 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)
%C A079309 a(n) is the number of terms of A031443 not exceeding 4^n. - _Vladimir Shevelev_, Oct 01 2010
%C A079309 Also the number of nonempty subsets of {1..2n} with median n, bisection of A361801. The version containing n is A001700 (bisected). Replacing 2n with 2n+1 and n with n+1 gives A006134. For mean instead of median we have A212352. - _Gus Wiseman_, Apr 16 2023
%H A079309 Vincenzo Librandi and Robert Israel, <a href="/A079309/b079309.txt">Table of n, a(n) for n = 1..1500</a> (terms 1..200 from Vincenzo Librandi).
%H A079309 A. Denise and R. Simion, <a href="http://dx.doi.org/10.1016/0012-365X(93)E0147-V">Two combinatorial statistics on Dyck paths</a>, Discrete Math., 137, 1995, 155-176.
%H A079309 Guo-Niu Han, <a href="/A196265/a196265.pdf">Enumeration of Standard Puzzles</a>, 2011. [Cached copy]
%H A079309 Guo-Niu Han, <a href="https://arxiv.org/abs/2006.14070">Enumeration of Standard Puzzles</a>, arXiv:2006.14070 [math.CO], 2020.
%H A079309 R. Witula, <a href="https://doi.org/10.1515/dema-2013-0418">Ramanujan type trigonometric formulas</a>, Demonstratio Mathematica, Vol. XLV, No. 4 (2012), 789-796. - From _N. J. A. Sloane_, Jan 01 2013
%F A079309 a(n) = (1/2)*(C(2, 1) + C(4, 2) + C(6, 3) + ... + C(2*n, n)) = A066796(n)/2. - _Vladeta Jovovic_, Feb 12 2003
%F A079309 G.f.: (1/sqrt(1 - 4*x) - 1)/(1 - x)/2. - _Vladeta Jovovic_, Feb 12 2003
%F A079309 Given g.f. A(x), then x * A(x - x^2) is g.f. of A024495. - _Michael Somos_, Feb 14 2006
%F A079309 a(n) = A066796(n)/2. - _Zerinvary Lajos_, Oct 25 2006
%F A079309 a(n) = Sum_{0 <= i <= j <= n} binomial(i+j, i). - _Benoit Cloitre_, Nov 25 2006
%F A079309 D-finite with recurrence n*a(n) + (-5*n+2)*a(n-1) + 2*(2*n-1)*a(n-2) = 0. - _R. J. Mathar_, Nov 30 2012
%F A079309 a(n) ~ 2^(2*n+1) / (3*sqrt(Pi*n)). - _Vaclav Kotesovec_, Feb 13 2014
%F A079309 a(n) = Sum_{k=0..n-1} A001700(k). - _Doug Bell_, Jun 23 2015
%F A079309 a(n) = -binomial(2*n+1, n)*hypergeom([1, n+3/2], [n+2], 4) - (i/sqrt(3) + 1)/2. - _Peter Luschny_, May 18 2018
%F A079309 From _Gus Wiseman_, Apr 18 2023: (Start)
%F A079309 a(n) = A024718(n) - 1.
%F A079309 a(n) = A231147(2n+1,n).
%F A079309 a(n) = A361801(2n) = A361801(2n+1). (End)
%F A079309 a(n) = Sum_{k=0..floor(n/2)} (-1)^k*binomial(2*n+2-k, n-2*k). - _Michael Weselcouch_, Jun 17 2025
%F A079309 a(n) = binomial(2*(1+n), n)*hypergeom([1, (1-n)/2, -n/2], [-2*(1+n), 3+n], 4). - _Stefano Spezia_, Jun 18 2025
%e A079309 a(4) = C(1,1) + C(3,2) + C(5,3) + C(7,4) = 1 + 3 + 10 + 35 = 49.
%e A079309 G.f. = x + 4*x^2 + 14*x^3 + 49*x^4 + 175*x^5 + 637*x^6 + 2353*x^7 + ...
%e A079309 From _Gus Wiseman_, Apr 16 2023: (Start)
%e A079309 The a(1) = 1 through a(3) = 14 subsets of {1..2n} with median n:
%e A079309 {1} {2} {3}
%e A079309 {1,3} {1,5}
%e A079309 {1,2,3} {2,4}
%e A079309 {1,2,4} {1,3,4}
%e A079309 {1,3,5}
%e A079309 {1,3,6}
%e A079309 {2,3,4}
%e A079309 {2,3,5}
%e A079309 {2,3,6}
%e A079309 {1,2,4,5}
%e A079309 {1,2,4,6}
%e A079309 {1,2,3,4,5}
%e A079309 {1,2,3,4,6}
%e A079309 {1,2,3,5,6}
%e A079309 (End)
%p A079309 a := n -> add(binomial(2*j, j)/2, j=1..n): seq(a(n), n=1..24); # _Zerinvary Lajos_, Oct 25 2006
%p A079309 a := n -> add(abs(binomial(-j, -2*j)), j=1..n): seq(a(n), n=1..24); # _Zerinvary Lajos_, Oct 03 2007
%p A079309 f:= gfun:-rectoproc({n*a(n) +(-5*n+2)*a(n-1) +2*(2*n-1)*a(n-2)=0,a(1)=1,a(2)=4},a(n),remember):
%p A079309 map(f, [$1..100]); # _Robert Israel_, Jun 24 2015
%t A079309 Rest[CoefficientList[Series[(1/Sqrt[1-4*x]-1)/(1-x)/2, {x, 0, 20}], x]] (* _Vaclav Kotesovec_, Feb 13 2014 *)
%t A079309 Accumulate[Table[Binomial[2n-1,n],{n,30}]] (* _Harvey P. Dale_, Jan 06 2021 *)
%o A079309 (PARI) {a(n) = sum(k=1, n, binomial(2*k - 1, k))}; /* _Michael Somos_, Feb 14 2006 */
%o A079309 (PARI) my(x='x+O('x^40)); Vec((1/sqrt(1-4*x)-1)/(1-x)/2) \\ _Altug Alkan_, Dec 24 2015
%Y A079309 Cf. A001700, A024495, A066796, A138341.
%Y A079309 Equals A024718(n) - 1.
%Y A079309 This is the even (or odd) bisection of A361801.
%Y A079309 A007318 counts subsets by length, A327481 by mean, A013580 by median.
%Y A079309 A359893 and A359901 count partitions by median.
%Y A079309 Cf. A000975, A057552, A231147, A361654, A361864, A362046.
%K A079309 easy,nonn
%O A079309 1,2
%A A079309 _Miklos Kristof_, Feb 10 2003
%E A079309 More terms from Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Feb 11 2003