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 A131428 #49 Mar 25 2025 12:49:53
%S A131428 1,1,3,9,27,83,263,857,2859,9723,33591,117571,416023,1485799,5348879,
%T A131428 19389689,70715339,259289579,955277399,3534526379,13128240839,
%U A131428 48932534039,182965127279,686119227299,2579808294647,9723892802903,36734706144303,139067101832007
%N A131428 a(n) = 2*C(n) - 1, where C(n) = A000108(n) are the Catalan numbers.
%C A131428 Starting (1, 3, 9, 27, 83, ...), = row sums of triangle A136522. - _Gary W. Adamson_, Jan 02 2008
%C A131428 Hankel transform is A171552. - _Paul Barry_, Dec 11 2009
%C A131428 Apparently, for n >= 1, the maximum peak height minus the maximum valley height summed over all Dyck n-paths (with max valley height deemed zero if no valleys). - _David Scambler_, Oct 05 2012
%C A131428 Apparently for n > 1 the number of fixed points in all Dyck (n-1)-paths. A fixed point occurs when a vertex of a Dyck k-path is also a vertex of the path U^kD^k. - _David Scambler_, May 01 2013
%H A131428 G. C. Greubel, <a href="/A131428/b131428.txt">Table of n, a(n) for n = 0..1000</a>
%F A131428 Right border of triangle A131429.
%F A131428 From _Emeric Deutsch_, Jul 25 2007: (Start)
%F A131428 a(n) = 2*binomial(2*n,n)/(n+1) - 1.
%F A131428 G.f.: (1-sqrt(1-4*x))/x - 1/(1-x). (End)
%F A131428 (1, 3, 9, 27, 83, ...) = row sums of A118976. - _Gary W. Adamson_, Aug 31 2007
%F A131428 Row sums of triangle A131428 starting (1, 3, 9, 27, 83, ...). - _Gary W. Adamson_, Aug 31 2007
%F A131428 Starting with offset 1 = Narayana transform (A001263) of [1,2,2,2,...]. - _Gary W. Adamson_, Jul 29 2011
%F A131428 D-finite with recurrence (n+1)*a(n) +2*(-2*n+1)*a(n-1) +3*(-n+1)=0. - _R. J. Mathar_, Nov 22 2024
%F A131428 a(n) = Sum_{k=0..n} ( binomial(n,k) - binomial(n,k-1) )^2 = Sum_{k=0..n} A080233(n,k)^2 = Sum_{k=0..n} A156644(n,k)^2. - _Seiichi Manyama_, Mar 25 2025
%e A131428 a(3) = 9 = 2*C(3) - 1 = 2*5 - 1, where C refers to the Catalan numbers, A000108.
%p A131428 seq(2*binomial(2*n,n)/(n+1)-1, n=0..25); # _Emeric Deutsch_, Jul 25 2007
%t A131428 2CatalanNumber[Range[0,25]]-1 (* _Harvey P. Dale_, Apr 17 2011 *)
%o A131428 (PARI) vector(25, n, n--; 2*binomial(2*n,n)/(n+1) - 1) \\ _G. C. Greubel_, Aug 12 2019
%o A131428 (Magma) [2*Catalan(n) -1: n in [0..25]]; // _G. C. Greubel_, Aug 12 2019
%o A131428 (Sage) [2*catalan_number(n) -1 for n in (0..25)] # _G. C. Greubel_, Aug 12 2019
%o A131428 (GAP) List([0..25], n-> 2*Binomial(2*n,n)/(n+1) - 1); # _G. C. Greubel_, Aug 12 2019
%Y A131428 Cf. A000108, A131427, A131429.
%Y A131428 Cf. A131428, A118976, A136522.
%Y A131428 Cf. A080233, A156644.
%K A131428 nonn
%O A131428 0,3
%A A131428 _Gary W. Adamson_, Jul 10 2007
%E A131428 More terms from _Emeric Deutsch_, Jul 25 2007