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 A362204 #15 Feb 20 2024 04:38:09
%S A362204 1,1,3,16,121,1176,13921,193978,3106881,56201176,1132709041,
%T A362204 25162197006,610668537073,16073212005436,455980333073721,
%U A362204 13868451147012946,450140785396634881,15529495879187075088,567427732311438658081,21889446540911251445206
%N A362204 Expansion of e.g.f. exp(x/sqrt(1-2*x)).
%F A362204 a(n) = n! * Sum_{k=0..n} (-2)^k * binomial(-(n-k)/2,k)/(n-k)! = n! * Sum_{k=0..n} 2^(n-k) * binomial(n-k/2-1,n-k)/k!.
%F A362204 From _Vaclav Kotesovec_, Feb 20 2024: (Start)
%F A362204 a(n) ~ 3^(-1/2) * 2^(n - 1/6) * exp(3*2^(-4/3)*n^(1/3) - n) * n^(n - 1/3) * (1 - 3/(16*(n/2)^(1/3))).
%F A362204 Recurrence (for n>5): (n-5)*a(n) = 3*(2*n^2 - 13*n + 16)*a(n-1) - (12*n^3 - 108*n^2 + 299*n - 259)*a(n-2) + (n-2)*(8*n^3 - 84*n^2 + 290*n - 327)*a(n-3) + (n-4)*(n-3)*(n-2)*a(n-4). (End)
%t A362204 Table[n! * Sum[2^(n-k) * Binomial[n-k/2-1,n-k]/k!, {k,0,n}], {n,0,20}] (* _Vaclav Kotesovec_, Feb 20 2024 *)
%t A362204 Join[{1, 1}, RecurrenceTable[{(-4 + n) (-3 + n) (-2 + n) a[-4 + n] + (-2 + n) (-327 + 290 n - 84 n^2 + 8 n^3) a[-3 + n] + (259 - 299 n + 108 n^2 - 12 n^3) a[-2 + n] + 3 (16 - 13 n + 2 n^2) a[-1 + n] + (5 - n) a[n] == 0, a[2] == 3, a[3] == 16, a[4] == 121, a[5] == 1176}, a, {n, 2, 20}]] (* _Vaclav Kotesovec_, Feb 20 2024 *)
%o A362204 (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x/sqrt(1-2*x))))
%Y A362204 Cf. A025168, A362163.
%K A362204 nonn,easy
%O A362204 0,3
%A A362204 _Seiichi Manyama_, Apr 11 2023