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 A126673 #25 Mar 05 2025 15:27:53
%S A126673 0,2,26,274,2844,30708,351504,4292496,55988640,779171040,11545476480,
%T A126673 181705299840,3029581820160,53376951801600,991337037465600,
%U A126673 19363464423475200,396915849843609600,8520964324004966400,191220598650009600000,4477883953203763200000,109242544826541772800000
%N A126673 Third diagonal of A126671.
%C A126673 It appears that a(n) = sum of invc(p) over all permutations p of {1,2,...,n}, where invc(p) is defined (by Carlitz) in the following way: express p in standard cycle form (i.e., cycles ordered by increasing smallest elements with each cycle written with its smallest element in the first position), then remove the parentheses and count the inversions in the obtained word. a(3)=2 because the six permutations 123,132,312,213,231 and 321 of {1,2,3} yield the words 123,123,132,123,123 and 132, respectively, having a total of 0+0+1+0+0+1 = 2 inversions. a(n) = Sum_{k>=0} k*A129178(n,k). - _Emeric Deutsch_, Oct 10 2007
%D A126673 L. Carlitz, Generalized Stirling numbers, Combinatorial Analysis Notes, Duke University, 1968, 1-7.
%H A126673 G. C. Greubel, <a href="/A126673/b126673.txt">Table of n, a(n) for n = 2..445</a>
%H A126673 M. Shattuck, <a href="http://www.emis.de/journals/INTEGERS/papers/f7/f7.Abstract.html">Parity theorems for statistics on permutations and Catalan words</a>, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 5, Paper A07, 2005.
%H A126673 N. J. A. Sloane, <a href="/A126671/a126671.txt">Notes on Carlo Wood's Polynomials</a>
%F A126673 a(n) = n! * (n*(n-5)/4 + 1 + 1/2 + ... + 1/n). - _Emeric Deutsch_, Oct 10 2007
%F A126673 E.g.f.: (2*x - 3*x^2 + 2*(1-x)^2 * log(1-x)) / (2*(-1+x)^3). - _G. C. Greubel_, May 05 2019
%F A126673 a(n) = 2 * Sum_{k>=1} k * A381529(n,k). - _Alois P. Heinz_, Feb 26 2025
%p A126673 seq(n!*(sum(1/k, k = 1 .. n)+(1/4)*n*(n-5)), n = 2 .. 21); # _Emeric Deutsch_, Oct 10 2007
%t A126673 Table[n!*(n*(n-5)/4 + HarmonicNumber[n]), {n,2,25}] (* _G. C. Greubel_, May 05 2019 *)
%o A126673 (PARI) my(x='x+O('x^30)); concat([0], Vec(serlaplace( (2*x - 3*x^2 + 2*(1-x)^2*log(1-x))/(2*(-1+x)^3) ))) \\ _G. C. Greubel_, May 05 2019
%o A126673 (Magma) [Factorial(n)*(n*(n-5)/4 + HarmonicNumber(n)): n in [2..25]]; // _G. C. Greubel_, May 05 2019
%o A126673 (Sage) [factorial(n)*(n*(n-5)/4 + harmonic_number(n)) for n in (2..25)] # _G. C. Greubel_, May 05 2019
%Y A126673 Cf. A129178, A381529.
%K A126673 nonn
%O A126673 2,2
%A A126673 _N. J. A. Sloane_ and Carlo Wood (carlo(AT)alinoe.com), Feb 13 2007