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 A115178 #26 Sep 10 2024 20:47:34
%S A115178 1,0,1,1,2,4,7,15,29,61,126,266,566,1212,2619,5685,12419,27247,60049,
%T A115178 132847,294931,656877,1467258,3286218,7378240,16603458,37441990,
%U A115178 84599854,191501532,434224404,986161959,2243009869,5108859821
%N A115178 Expansion of c(x^2+x^3), c(x) the g.f. of A000108.
%C A115178 Diagonal sums of number triangle A117434.
%C A115178 a(n) = number of Motzkin n-paths (A001006) in which every flatstep (F) is followed by a downstep (D). For example, a(5)=4 counts UDUFD, UFDUD, UUDFD, UUFDD. - _David Callan_, Jun 07 2006
%C A115178 a(n) = number of lattice paths in the first quadrant from (0,0) to (n,0) using only steps U1=(1,1), U2=(2,1) and D=(1,-1). E.g. a(6)=7 because we have U1DU1DU1D, U1U1U1DDD, U1U1DU1DD, U1DU1U1DD, U1U1DDU1D, U2DU2D and U2U2DD. - _José Luis Ramírez Ramírez_, May 27 2013
%H A115178 G. C. Greubel, <a href="/A115178/b115178.txt">Table of n, a(n) for n = 0..1000</a>
%F A115178 a(n) = Sum_{k=0..floor(n/2)} C(k)*C(k,n-2k).
%F A115178 D-finite with recurrence (n+2)*a(n) +(n+2)*a(n-1) +4*(1-n)*a(n-2) +2*(7-4*n)*a(n-3) +2*(5-2*n)*a(n-4)=0. - _R. J. Mathar_, Nov 15 2011
%F A115178 G.f. A(x) satisfies A(x) = 1 / (1 - x^2 / (1 - x / (1 - x^2 * A(x)))). - _Michael Somos_, May 12 2012
%F A115178 G.f.: (1-sqrt(1-4*z^2*(1+z)))/(2*z^2*(1+z)). - _José Luis Ramírez Ramírez_, May 27 2013
%F A115178 a(n) ~ sqrt(3 - 1/9*(-2 + (19-3*sqrt(33))^(1/3) + (19+3*sqrt(33))^(1/3))^2) * (((-2 + (19-3*sqrt(33))^(1/3) + (19+3*sqrt(33))^(1/3)) * (4 + (19-3*sqrt(33))^(1/3) + (19+3*sqrt(33))^(1/3)))/9)^n /(n^(3/2)*sqrt(Pi)). - _Vaclav Kotesovec_, Sep 16 2013
%e A115178 1 + x^2 + x^3 + 2*x^4 + 4*x^5 + 7*x^6 + 15*x^7 + 29*x^8 + 61*x^9 + ...
%t A115178 Table[Sum[Binomial[k, n - 2*k]*CatalanNumber[k], {k, 0, Floor[n/2]}], {n, 0, 50}] (* _G. C. Greubel_, Feb 03 2017 *)
%o A115178 (PARI) {a(n) = local(A); A = O(x^0); for( k=0, n\5, A = 1 / (1 - x^2 / (1 - x / (1 - x^2 * A)))); polcoeff( A, n)} /* _Michael Somos_, May 12 2012 */
%Y A115178 Cf. A007477.
%K A115178 easy,nonn
%O A115178 0,5
%A A115178 _Paul Barry_, Mar 14 2006