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%I #47 Jul 19 2017 21:08:41
%S 1,2,6,16,36,71,127,211,331,496,716,1002,1366,1821,2381,3061,3877,
%T 4846,5986,7316,8856,10627,12651,14951,17551,20476,23752,27406,31466,
%U 35961,40921,46377,52361,58906,66046,73816,82252,91391,101271,111931,123411,135752
%N a(n) = 1 + (6 + (11 + (6 + n)*n)*n)*n/24.
%C From _Gary W. Adamson_, Jul 31 2010: (Start)
%C Equals (1, 2, 3, 4, 5, ...) convolved with (1, 0, 3, 6, 10, 15, ...).
%C Example: a(4) = 36 = (5, 4, 3, 2, 1) dot (1, 0, 3, 6, 10) = (5 + 0 + 9 + 12 + 10). (End)
%C Also the number of permutations of length n that can be sorted by a single block interchange (in the sense of Christie). - _Vincent Vatter_, Aug 21 2013
%H Vincenzo Librandi, <a href="/A145126/b145126.txt">Table of n, a(n) for n = 0..1000</a>
%H D. A. Christie, <a href="http://dx.doi.org/10.1016/S0020-0190(96)00155-X">Sorting Permutations by Block-Interchanges</a>, Inf. Process. Lett. 60 (1996), 165-169.
%H Cheyne Homberger, <a href="https://arxiv.org/abs/1410.2657">Patterns in Permutations and Involutions: A Structural and Enumerative Approach</a>, arXiv preprint 1410.2657 [math.CO], 2014.
%H C. Homberger and V. Vatter, <a href="http://www.math.ufl.edu/~vatter/publications/poly-classes/">On the effective and automatic enumeration of polynomial permutation classes</a>. [Broken link]
%H C. Homberger, V. Vatter, <a href="https://arxiv.org/abs/1308.4946">On the effective and automatic enumeration of polynomial permutation classes</a>, arXiv preprint arXiv:1308.4946 [math.CO], 2013-2015.
%H Luis Manuel Rivera, <a href="https://arxiv.org/abs/1406.3081">Integer sequences and k-commuting permutations</a>, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F G.f.: (x^4-4*x^3+6*x^2-3*x+1) / (1-x)^5.
%F a(n) = C(n+3,4)+1. - _Zerinvary Lajos_, Mar 24 2009
%p a:= n-> 1+ (6+ (11+ (6+ n) *n) *n) *n/24: seq(a(n), n=0..40);
%p # second Maple program:
%p with(combinat): seq(binomial(n+3, 4)+1, n=0..40); # _Zerinvary Lajos_, Mar 24 2009
%t a=b=s=0;lst={a};Do[a+=n;b+=a;s+=b;AppendTo[lst,s],{n,6!}];lst+1 (* _Vladimir Joseph Stephan Orlovsky_, Jun 14 2009 *)
%t CoefficientList[Series[(x^4 - 4 x^3 + 6 x^2 - 3 x + 1) / (1 - x)^5, {x, 0, 50}], x] (* _Vincenzo Librandi_, Jun 06 2013 *)
%o (PARI) Vec((x^4-4*x^3+6*x^2-3*x+1)/(1-x)^5 + O(x^50)) \\ _Altug Alkan_, Nov 24 2015
%Y 5th row of A145153. See row 5 of A145140/A145141 for rational coefficients and A145142 for 24 * coefficients of polynomial.
%K nonn,easy
%O 0,2
%A _Alois P. Heinz_, Oct 03 2008
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