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

1, 6, 25, 132, 824, 5932, 48444, 442916, 4484524, 49828044, 602919332, 7892762164, 111156400476, 1675896499484, 26934050884564, 459674468429892, 8302870086014924, 158242935756990316, 3173649989348528004, 66813683986284800084, 1473241731897579841852

Offset: 1

Author

Benoit Cloitre, Apr 24 2003

Keywords

• nonn

Comments

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.)

Examples

			r(10) = frac(10*e) + 1/2 + 2;
r(12) = frac(12*e) - 1/2 + 3;
r(15) = frac(15*e) + 3;
r(18) = frac(18*e) - 1/2 + 4.
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..448

Crossrefs

Cf. A007808, A074143, A082425, A082427, A082428, A383436, A383437.

Cf. A001339.

Programs

Mathematica

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 *)

Formula

For n >= 2, a(n) = ceiling(n*(19/2 - 4*e)*n!).

From Seiichi Manyama, Apr 27 2025: (Start)

E.g.f.: -4 - 3*x/2 + (-19*x/2 + 4*exp(x))/(1-x)^2.

a(n) = -19*n/2 * n! + 4 * Sum_{k=0..n} (k+1)! * binomial(n,k) for n > 1.

a(n) = (n^2 * a(n-1) - 4)/(n-1) for n > 2.

a(n) = (n+2) * a(n-1) - (n-1) * a(n-2) for n > 3. (End)