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 A346943 #26 Apr 02 2025 09:43:26
%S A346943 1,1,7,19,159,729,7407,48231,581535,4922325,68891175,718638075,
%T A346943 11465661375,142257791025,2550046679775,36691916525775,
%U A346943 730304613424575,11958031070311725,261722208861516375,4805774015579971875,114729101737416849375,2334996696935363855625
%N A346943 a(n) = a(n-1) + n*(n+1)*a(n-2) with a(0)=1, a(1)=1.
%C A346943 From _Peter Bala_, Dec 09 2024: (Start)
%C A346943 b(n) := A000246(n+2) = (n+2)!/2^(n+1) * binomial(n+1, floor((n+1)/2)) satisfies the same second-order recurrence as a(n) with the initial conditions b(0) = 1 and b(1) = 3. This leads to the finite continued fraction a(n)/b(n) = 1/(1 + 2/(1 + 6/(1 + ... + n*(n+1)/1). Letting n tend to infinity gives the continued fraction representation 1/(1 + 2/(1 + 6/(1 + ... + n*(n+1)/(1 + ...) = Pi/2 - 1, due to Euler - see paragraph 31, p. 48. (End)
%H A346943 Leonhard Euler, <a href="https://scholarlycommons.pacific.edu/euler-works/123/">E123: De fractionibus continuis observationes</a>, originally published in Commentarii academiae scientiarum Petropolitanae 11, 1750, pp. 32-81, reprinted in Opera Omnia: Series 1, Volume 14, 291-349.
%F A346943 a(n) ~ n! * (Pi - 2) * n^(3/2) / sqrt(2*Pi).
%F A346943 a(n) ~ (Pi - 2) * n^(n+2) / exp(n).
%F A346943 E.g.f. A(x) satisfies the differential equation -6*A(x) - (6*x + 1)*A'(x) + (1 - x^2)*A''(x) = 0, A(0)=1, A'(0)=1.
%F A346943 E.g.f.: (-2 + Pi + 2*Pi*x + 4*sqrt(1-x^2) + 2*x*(-2+sqrt(1-x^2)) - 4*(1+2*x) * arcsin(sqrt(1-x)/sqrt(2))) / (2*(1-x)^(5/2) * (1+x)^(3/2)).
%t A346943 RecurrenceTable[{a[n] == a[n-1] + n*(n+1)*a[n-2], a[0]==1, a[1]==1}, a, {n,0,20}]
%t A346943 nmax = 20; CoefficientList[Series[(-2 + Pi + 2*Pi*x + 4*Sqrt[1 - x^2] + 2*x*(-2 + Sqrt[1 - x^2]) - 4*(1 + 2*x) * ArcSin[Sqrt[1 - x]/Sqrt[2]]) / (2*(1 - x)^(5/2) * (1 + x)^(3/2)), {x, 0, nmax}], x] * Range[0, nmax]!
%Y A346943 Cf. A000246, A000932, A024167.
%K A346943 nonn,easy
%O A346943 0,3
%A A346943 _Vaclav Kotesovec_, Aug 08 2021, following a suggestion from _John M. Campbell_