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 A122057 #40 Jan 04 2025 16:32:13
%S A122057 0,2,14,94,684,5508,49104,482256,5185440,60668640,767940480,
%T A122057 10462227840,152698210560,2377651449600,39350097561600,
%U A122057 689874448435200,12773427499929600,249097496204390400,5103595024496640000,109608397522606080000,2462475687669043200000
%N A122057 a(n) = (n+1)! * (H(n+1) - H(2)), where H(n) are the harmonic numbers.
%C A122057 Former title (corrected): A Legendre-based recurrence sequence. Let b(n) = ((4*n+2)*x -(2*n+1) )/(n+1)*b(n-1) - (n/(n+1))*b(n-2), where x=1, then a(n) = (n+1)!*b(n)/6. - _G. C. Greubel_, Oct 03 2019
%D A122057 Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964, 9th Printing (1970), pp. 782
%H A122057 G. C. Greubel, <a href="/A122057/b122057.txt">Table of n, a(n) for n = 1..445</a>
%H A122057 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%F A122057 Let b(n) = ((-2*n-1) +(4*n+2)*x)/(n+1)*b(n-1) - (n/(n+1))*b(n-2) with x=1, then a(n) = b(n)*(n+1)!/6.
%F A122057 a(n) = (n+1)! * Sum_{k=3..n+1} 1/k. - _Gary Detlefs_, Jul 15 2010
%F A122057 a(n) = 2*A001711(n-2) for n >= 2. - _Pontus von Brömssen_, Jan 04 2025
%p A122057 a:=n-> (n+1)!*add(1/k,k=3..n+1): seq(a(n),n=1..30); # _Gary Detlefs_, Jul 15 2010
%t A122057 x=1; b[1]:=0; b[2]:=2; b[n_]:= b[n]= ((-2*n-1) +(4*n+2)*x)/(n+1)*b[n-1] - (n/(n+1))*b[n-2]; Table[b[n]*(n+1)!/6, {n,30}]
%t A122057 Table[(n+1)!*(HarmonicNumber[n+1] - 3/2), {n,30}] (* _G. C. Greubel_, Oct 03 2019 *)
%o A122057 (PARI) vector(30, n, (n+1)!*(sum(k=1,n+1, 1/k) - 3/2) ) \\ _G. C. Greubel_, Oct 03 2019
%o A122057 (Magma) [Factorial(n+1)*(HarmonicNumber(n+1) - 3/2): n in [1..30]]; // _G. C. Greubel_, Oct 03 2019
%o A122057 (Sage) [factorial(n+1)*(harmonic_number(n+1) - 3/2) for n in (1..30)] # _G. C. Greubel_, Oct 03 2019
%o A122057 (GAP) List([1..30], n-> Factorial(n+1)*(Sum([1..n+1], k-> 1/k) - 3/2) ); # _G. C. Greubel_, Oct 03 2019
%Y A122057 Cf. A001008, A002805, A001705, A001711.
%K A122057 nonn
%O A122057 1,2
%A A122057 _Roger L. Bagula_, Sep 14 2006
%E A122057 If all terms are really negative, sequence should probably be negated. - _N. J. A. Sloane_, Oct 01 2006
%E A122057 Negated terms and edited by _G. C. Greubel_, Oct 03 2019