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 A182541 #88 Sep 05 2024 08:07:59
%S A182541 1,4,19,107,702,5274,44712,422568,4407120,50292720,623471040,
%T A182541 8344624320,119938250880,1842662908800,30136443724800,522780938265600,
%U A182541 9587900602828800,185371298306611200,3768248516336640000,80349669847157760000,1793238207723325440000,41806479141525288960000
%N A182541 Coefficients in g.f. for certain marked mesh patterns.
%C A182541 See Kitaev and Remmel for precise definition.
%C A182541 The listed terms a(3)-a(10) of this sequence can be produced by the formula (n-1)!*(H(n-1)-1/2)/2, where H(n) = A001008(n)/A002805(n) is the n-th harmonic number. - _Gary Detlefs_, May 28 2012
%C A182541 a(n) is also the number of nonzero elements left in the matrix where all the rows consist of permutations of 11...n after we delete for each element with the value of 'k' k elements of this type, and repeat this operation until no more elements with the value of k can be deleted. The whole operation should be done for all the values of k from 1 to n. - _Anton Zakharov_, Jun 28 2016
%H A182541 Joerg Arndt, <a href="/A182541/b182541.txt">Table of n, a(n) for n = 3..101</a>
%H A182541 Sergey Kitaev and Jeffrey Remmel, <a href="http://arxiv.org/abs/1201.1323">Simple marked mesh patterns</a>, arXiv preprint arXiv:1201.1323 [math.CO], 2012.
%H A182541 Anton Zakharov, <a href="/A182541/a182541_2.pdf">Matrix-related sequences</a>
%F A182541 a(n) = A001710(n+1) * (1 + Sum_{k=2..n} 1/(k+1) ). - _Anton Zakharov_, Jun 28 2016
%F A182541 a(n) ~ sqrt(Pi/2)*exp(-n)*n^(n-1/2)*log(n). - _Ilya Gutkovskiy_, Jul 12 2016
%F A182541 From _Pedro Caceres_, Apr 19 2019: (Start)
%F A182541 a(n) = (n-3)! * Sum_{i=1..n-2} (Sum_{j=1..i} (i/j)).
%F A182541 a(n) = (1/4) * (n-1)! * (2*harmonic(n-1)-1). (End)
%F A182541 a(n) = (-(n-1)! + 2 * |Stirling1(n,2)|)/4. - _Seiichi Manyama_, Sep 05 2024
%e A182541 a(1) corresponds to the 1 X 2 matrix 11 -> 1 element is left and there are no more ones to delete => n(1) = 1. a(2) corresponds to the 3 X 3 matrix 112 121 211 -> 102 120 210 -> 102 100 010 only 4 nonzero elements are left and a(2) = 4 = 3 + 3/3. a(3) = 12 + 12/3 + 12/4 = 19 = 19 nonzero elements left in the 4 X 12 matrix after the deletion for each element with the value of 1 one element with the value of 1, for every element with the value of 2 - two elements with the value of 2 and for each element with the value of 3 - three elements with the value of 3). - _Anton Zakharov_, Jun 28 2016
%t A182541 Table[Numerator[(n+1)!/2] *(1 + Sum[1/(k+1), {k, 2, n}]),{n, 1, 22}] (* _Indranil Ghosh_, Mar 12 2017 *)
%o A182541 (PARI) for(n=1, 22, print1(numerator((n + 1)!/2) * (1 + sum(k=2, n, 1/(k+1))),", ")) \\ _Indranil Ghosh_, Mar 12 2017
%Y A182541 Cf. A001710, A307642.
%K A182541 nonn
%O A182541 3,2
%A A182541 _N. J. A. Sloane_, May 04 2012
%E A182541 More terms from _Anton Zakharov_, Jun 28 2016