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 A323770 #17 Mar 06 2021 03:34:06
%S A323770 0,1,5,30,214,1775,16791,178360,2101100,27172269,382566025,5823044546,
%T A323770 95253119490,1666020561595,31019392831259,612430207741500,
%U A323770 12778091116288216,280893425932078745,6487870112636577165,157066777096248548134,3976727555939887035950,105087648979005066820551
%N A323770 Expansion of e.g.f. x*(2 - x)*exp(x/(1 - x))/(2*(1 - x)^2).
%H A323770 G. C. Greubel, <a href="/A323770/b323770.txt">Table of n, a(n) for n = 0..430</a>
%F A323770 a(n) = Sum_{k=0..n} binomial(n-1,k-1)*A000217(k)*n!/k!.
%F A323770 a(n) ~ n^(n + 3/4) / (2^(3/2) * exp(n - 2*sqrt(n) + 1/2)). - _Vaclav Kotesovec_, Jan 27 2019
%F A323770 a(n) = n!*Hypergeometric2F2([1-n, 3], [1, 2], -1). - _G. C. Greubel_, Mar 05 2021
%p A323770 seq(n!*coeff(series(x*(2-x)*exp(x/(1-x))/(2*(1-x)^2),x=0,22),x,n),n=0..21); # _Paolo P. Lava_, Jan 29 2019
%t A323770 nmax = 21; CoefficientList[Series[x (2 - x) Exp[x/(1 - x)]/(2 (1 - x)^2), {x, 0, nmax}], x] Range[0, nmax]!
%t A323770 Table[Sum[Binomial[n - 1, k - 1] Binomial[k + 1, 2] n!/k!, {k, 0, n}], {n, 0, 21}]
%o A323770 (Sage) [0]+[sum(binomial(n-1,k)*binomial(k+2,2)*factorial(n)/factorial(k+1) for k in (0..n-1)) for n in [1..20]] # _G. C. Greubel_, Mar 05 2021
%o A323770 (Magma) [n eq 0 select 0 else (&+[Binomial(n-1,k)*Binomial(k+2,2)* Factorial(n)/Factorial(k+1): k in [0..n-1]]): n in [0..20]]; // _G. C. Greubel_, Mar 05 2021
%Y A323770 Cf. A000217, A052852, A103194.
%K A323770 nonn
%O A323770 0,3
%A A323770 _Ilya Gutkovskiy_, Jan 27 2019