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A082430 a(1)=1; for n > 1, a(n) = n*(a(n-1) + a(n-2) + ... + a(2) + a(1)) + 4.

%I A082430 #28 Apr 27 2025 11:50:45
%S A082430 1,6,25,132,824,5932,48444,442916,4484524,49828044,602919332,
%T A082430 7892762164,111156400476,1675896499484,26934050884564,459674468429892,
%U A082430 8302870086014924,158242935756990316,3173649989348528004,66813683986284800084,1473241731897579841852
%N A082430 a(1)=1; for n > 1, a(n) = n*(a(n-1) + a(n-2) + ... + a(2) + a(1)) + 4.
%C A082430 More generally, if m is an integer and a(1)=1, a(n) = n*(a(n-1) + a(n-2) + ... + a(2) + a(1)) + m then a(n) has a closed form formula as a(n) = floor/ceiling(n*r(m)*n!) where r(m) = frac(e*m) + 0 or + 1/2 or -1/2 + integer. (See Example section.)
%H A082430 Harvey P. Dale, <a href="/A082430/b082430.txt">Table of n, a(n) for n = 1..448</a>
%F A082430 For n >= 2, a(n) = ceiling(n*(19/2 - 4*e)*n!).
%F A082430 From _Seiichi Manyama_, Apr 27 2025: (Start)
%F A082430 E.g.f.: -4 - 3*x/2 + (-19*x/2 + 4*exp(x))/(1-x)^2.
%F A082430 a(n) = -19*n/2 * n! + 4 * Sum_{k=0..n} (k+1)! * binomial(n,k) for n > 1.
%F A082430 a(n) = (n^2 * a(n-1) - 4)/(n-1) for n > 2.
%F A082430 a(n) = (n+2) * a(n-1) - (n-1) * a(n-2) for n > 3. (End)
%e A082430 r(10) = frac(10*e) + 1/2 + 2;
%e A082430 r(12) = frac(12*e) - 1/2 + 3;
%e A082430 r(15) = frac(15*e) + 3;
%e A082430 r(18) = frac(18*e) - 1/2 + 4.
%t A082430 nxt[{n_,t_,a_}]:=Module[{c=t(n+1)+4},{n+1,t+c,c}]; NestList[nxt,{1,1,1},20][[;;,3]] (* _Harvey P. Dale_, Mar 28 2024 *)
%Y A082430 Cf. A007808, A074143, A082425, A082427, A082428, A383436, A383437.
%Y A082430 Cf. A001339.
%K A082430 nonn
%O A082430 1,2
%A A082430 _Benoit Cloitre_, Apr 24 2003