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A324356 Total number of occurrences of 6 in the (signed) displacement sets of all permutations of [n+6] divided by 6!.

%I A324356 #9 Jul 06 2021 17:14:51
%S A324356 0,1,13,148,1707,20686,266321,3652608,53339831,827870338,13624599309,
%T A324356 237169578724,4356110013107,84220077081414,1710164008931657,
%U A324356 36396070427846536,810244122520224111,18833465673721387018,456310533309915775301,11505888654389005045548
%N A324356 Total number of occurrences of 6 in the (signed) displacement sets of all permutations of [n+6] divided by 6!.
%H A324356 Alois P. Heinz, <a href="/A324356/b324356.txt">Table of n, a(n) for n = 0..445</a>
%H A324356 Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation">Permutation</a>
%F A324356 E.g.f.: (1-exp(-x))/(1-x)^7.
%F A324356 a(n) = -1/6! * Sum_{j=1..n} (-1)^j * binomial(n,j) * (n+6-j)!.
%F A324356 a(n) = A306234(n+6,6).
%p A324356 a:= n-> (k-> -add((-1)^j*binomial(n, j)*(n+k-j)!, j=1..n)/k!)(6):
%p A324356 seq(a(n), n=0..23);
%t A324356 With[{nn=20},CoefficientList[Series[(1-Exp[-x])/(1-x)^7,{x,0,nn}],x] Range[ 0,nn]!] (* _Harvey P. Dale_, Jul 06 2021 *)
%Y A324356 Column k=6 of A324362.
%Y A324356 Cf. A306234.
%K A324356 nonn
%O A324356 0,3
%A A324356 _Alois P. Heinz_, Feb 23 2019