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 A380042 #11 Jan 11 2025 10:27:39
%S A380042 1,1,5,48,697,13640,336771,10053778,352334753,14183529480,
%T A380042 645073504435,32715111226886,1830671281889649,112049330303532388,
%U A380042 7446824171300128811,534068807341887943770,41111698162393482004801,3381089519620006418116976,295869084136630532211207843
%N A380042 E.g.f. A(x) satisfies A(x) = 1/sqrt( 1 - 2*x*exp(x*A(x)^2) ).
%F A380042 E.g.f.: sqrt( (1/x) * Series_Reversion(x/(1 + 2*x*exp(x))) ).
%F A380042 a(n) = (n!/(2*n+1)) * Sum_{k=0..n} 2^k * k^(n-k) * binomial(n+1/2,k)/(n-k)!.
%o A380042 (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(sqrt(serreverse(x/(1+2*x*exp(x)))/x)))
%o A380042 (PARI) a(n) = n!*sum(k=0, n, 2^k*k^(n-k)*binomial(n+1/2, k)/(n-k)!)/(2*n+1);
%Y A380042 Cf. A380015, A380035.
%Y A380042 Cf. A161633, A380043.
%Y A380042 Cf. A201470.
%K A380042 nonn
%O A380042 0,3
%A A380042 _Seiichi Manyama_, Jan 10 2025