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A328088 a(n) = Sum_{k=4..n} ( binomial(n,k)*(k-2)*(2^k-2*k-2) ) - (2^n-n-1).

%I A328088 #22 Oct 20 2019 14:22:22
%S A328088 1,94,683,3520,15461,61826,232543,838276,2930585,10014406,33633299,
%T A328088 111448904,365403853,1187875594,3834883271,12309375244,39320806145,
%U A328088 125090127950,396537120379,1253145232336,3949433330741,12416933938834,38953666980143,121962851990420,381179210953321,1189376848680406,3705576521235683
%N A328088 a(n) = Sum_{k=4..n} ( binomial(n,k)*(k-2)*(2^k-2*k-2) ) - (2^n-n-1).
%H A328088 Robert Israel, <a href="/A328088/b328088.txt">Table of n, a(n) for n = 4..2087</a>
%H A328088 J. B. Remmel et al., <a href="https://doi.org/10.1016/j.disc.2019.05.026">The combinatorial properties of the Benoumhani polynomials for the Whitney numbers of Dowling lattices</a>, Discrete Math., 342 (2019), 2966-2983. See page 2981, formula for coefficient of m in B_{m+1}(n,n-2).
%H A328088 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (14,-82,260,-481,518,-300,72).
%F A328088 a(n) = 3^(n-1)*(2*n-6) + 2^(n-1)*(-n^2+n+6) - n - 1. - _Robert Israel_, Oct 18 2019
%F A328088 From _Colin Barker_, Oct 19 2019: (Start)
%F A328088 G.f.: x^4*(1 + 80*x - 551*x^2 + 1406*x^3 - 1772*x^4 + 1128*x^5 - 288*x^6) / ((1 - x)^2*(1 - 2*x)^3*(1 - 3*x)^2).
%F A328088 a(n) = -1 + 3*2^n - 2*3^n + (1/6)*(-6 + 3*2^n + 4*3^n)*n - 2^(-1+n)*n^2  for n>3.
%F A328088 a(n) = 14*a(n-1) - 82*a(n-2) + 260*a(n-3) - 481*a(n-4) + 518*a(n-5) - 300*a(n-6) + 72*a(n-7) for n>10.
%F A328088 (End)
%F A328088 E.g.f.: x^2/2 + 2*x^3/3 + exp(2*x)*(3 - 2*x^2 + (-3 + x)*cosh(x) + (-1 + 3*x)*sinh(x)). - _Stefano Spezia_, Oct 19 2019
%p A328088 f:= n -> 3^(n-1)*(2*n-6) + 2^(n-1)*(-n^2+n+6) - n - 1:
%p A328088 map(f, [$4..40]); # _Robert Israel_, Oct 18 2019
%o A328088 (PARI) Vec(x^4*(1 + 80*x - 551*x^2 + 1406*x^3 - 1772*x^4 + 1128*x^5 - 288*x^6) / ((1 - x)^2*(1 - 2*x)^3*(1 - 3*x)^2) + O(x^40)) \\ _Colin Barker_, Oct 19 2019
%Y A328088 For the constant term see A000460.
%K A328088 nonn,easy
%O A328088 4,2
%A A328088 _N. J. A. Sloane_, Oct 17 2019