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 A115197 #10 Apr 22 2025 11:06:50 %S A115197 1,2,7,32,169,974,5947,37820,247885,1662890,11362399,78806936, %T A115197 553386097,3926523782,28108587139,202764451700,1472446595221, %U A115197 10755543924578,78973277044903,582558618222416,4315238786662585 %N A115197 Convolution of generalized Catalan numbers A064062 (called C(n;2)). %C A115197 Row sums of triangle A115193, called C(1,2). %C A115197 The o.g.f. given below follows from the Riordan matrix structure of the triangle A115193. See the o.g.f. for the row polynomials of A115193. %H A115197 Joseph Abate and Ward Whitt, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Whitt/whitt6.html">Brownian Motion and the Generalized Catalan Numbers</a>, J. Int. Seq. 14 (2011) # 11.2.6, after theorem 5. %H A115197 Owen John Levens, Joel Brewster Lewis, and Bridget Eileen Tenner, <a href="https://arxiv.org/abs/2504.13108">Global patterns in signed permutations</a>, arXiv:2504.13108 [math.CO], 2025. See p. 19. %F A115197 a(n)= sum(A115193(n,m),m=0..n), n>=0. %F A115197 G.f.: ((1+2*x*c(2*x))/(1+x))^2 = ((1-2*x) + 6*x*c(2*x))/(1+x)^2, with the o.g.f. c(x) of Catalan numbers A000108. %F A115197 a(n)= sum(C(2;n-k)*C(2;k),k=0..n), n>=0, with C(2;n):= A064062(n). %F A115197 a(n)=4*A178792(n)-3*(n+1)*A064062(n+1) [From _Joseph Abate_, Jun 21 2010] %F A115197 n*a(n) +(-7*n+13)*a(n-1) +4*(-2*n+1)*a(n-2)=0. - _R. J. Mathar_, Aug 09 2017 %K A115197 nonn %O A115197 0,2 %A A115197 _Wolfdieter Lang_, Feb 23 2006