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A228997 Total sum of the 7th powers of lengths of ascending runs in all permutations of [n].

%I A228997 #14 Jul 31 2021 17:15:59
%S A228997 0,1,130,2706,32226,315684,2961498,28544040,291590754,3194874900,
%T A228997 37656861354,477018980928,6477756701010,94006723773564,
%U A228997 1453236561824250,23855684885059944,414605141516228418,7607828522859788580,147012653519046471114,2984603478905797978320
%N A228997 Total sum of the 7th powers of lengths of ascending runs in all permutations of [n].
%H A228997 Alois P. Heinz, <a href="/A228997/b228997.txt">Table of n, a(n) for n = 0..200</a>
%F A228997 E.g.f.: (exp(x)*(42*x^5+210*x^4+280*x^3+126*x-126)+x+126)/(x-1)^2.
%F A228997 a(n) ~ n! * (532*exp(1)+127)*n. - _Vaclav Kotesovec_, Sep 12 2013
%p A228997 a:= proc(n) option remember; `if`(n<4, [0, 1, 130, 2706][n+1],
%p A228997       ((16*n^3-38*n^2-16*n+278) *a(n-1)
%p A228997       -(8*n^4-3*n^3-101*n^2+623*n-512) *a(n-2)
%p A228997       +2*(n-2)*(8*n^3-32*n^2+134*n-95) *a(n-3)
%p A228997       -(n-2)*(n-3)*(8*n^2-37*n+44) *a(n-4)) /(8*n^2-27*n+24))
%p A228997     end:
%p A228997 seq(a(n), n=0..30);
%t A228997 With[{nn=20},CoefficientList[Series[(Exp[x](42x^5+210x^4+280x^3+126x-126)+x+126)/(x-1)^2,{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, Jul 31 2021 *)
%Y A228997 Column k=7 of A229001.
%K A228997 nonn
%O A228997 0,3
%A A228997 _Alois P. Heinz_, Sep 10 2013