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 A262607 #19 Jan 08 2025 11:00:28 %S A262607 1,3,11,47,219,1075,5459,28383,150131,804515,4355163,23768079, %T A262607 130572363,721247571,4002344355,22296869823,124633584099,698707769923, %U A262607 3927060020651,22121780745711,124865811262139,706065855417203,3998950848888051 %N A262607 Sum_{k=0..n} ((k+1)*binomial(n+1,k)*binomial(2*n-k,n))/(n+1). %H A262607 D. Drake, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Drake/drake.html">Bijections from Weighted Dyck Paths to Schröder Paths</a>, J. Int. Seq. 13 (2010) # 10.9.2, Table 1. %F A262607 G.f.: (-2*x^2+7*x-1)/(2*x*sqrt(x^2-6*x+1))+1/(2*x)-1. %F A262607 G.f. satisfies -A'(x)/A(x)+A'(x)/x, where A(x)/x is g.f. of A155069 %F A262607 -(n+1)*(2*n^2+5*n-6)*a(n) +6*(2*n^3+6*n^2-11*n+4)*a(n-1) -(n-2)*(2*n^2+9*n+1)*a(n-2)=0. - _R. J. Mathar_, Jul 21 2017 %F A262607 a(n) ~ (1 + sqrt(2))^(2*n) / (2^(5/4) * sqrt(Pi*n)). - _Vaclav Kotesovec_, Jul 10 2021 %t A262607 Table[Sum[(k + 1) Binomial[n + 1, k] Binomial[2 n - k, n]/(n + 1), {k, %t A262607 0, n}], {n, 0, 22}] (* _Michael De Vlieger_, Sep 26 2015 *) %o A262607 (Maxima) %o A262607 A(x):=x*(3-x-sqrt(1-6*x+x^2))/2; %o A262607 taylor(-diff(A(x),x)/A(x)+diff(A(x),x,1)/x,x,0,27); %Y A262607 Cf. A155069. %K A262607 nonn %O A262607 0,2 %A A262607 _Vladimir Kruchinin_, Sep 26 2015