A144495 Row 2 of array in A144502.

2, 7, 30, 155, 946, 6687, 53822, 486355, 4877250, 53759351, 646098622, 8409146187, 117836551730, 1768850337295, 28318532194206, 481652022466307, 8673291031865602, 164849403644999655, 3297954931572397790, 69274457019123638011, 1524368720086682440242

Offset

0,1

Links

Robert Israel, Table of n, a(n) for n = 0..447

Formula

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

a(n) = (1/2) * (floor((n+1)*(n+1)!*e) + floor(n*n!*e)). [Gary Detlefs, Jun 06 2010]

a(n) = (1/2) * ( A001339(n) + A001339(n+1) ). [Gary Detlefs, Jun 06 2010]

a(n) = (1/2) * (3 + n + (1 + 3*n + n^2) * A000522(n)). - Gerry Martens, Oct 02 2016

a(n) = ((4+3*n)*a(n-1) - (n+3)*(n-1)*a(n-2) + (n-1)*(n-2)*a(n-3))/2. - Robert Israel, Oct 02 2016

From Peter Bala, May 27 2022: (Start)

a(n) = (1/2)*(A000522(n+2) - A000522(n)).

a(n) = (1/2)*Sum_{k = 0..n} binomial(n,k)*(k+4)*(k+1)!; binomial transform of A038720(n+1).

a(n) = (1/2)*e*Integral_{x >= 1} x^n*(x^2 - 1)*exp(-x).

a(2*n) is even and a(2*n+1) is odd. More generally, a(n+k) == a(n) (mod k) for all n and k. It follows that for each positive integer k, the sequence obtained by reducing a(n) modulo k is periodic, with the exact period dividing k. Various divisibility properties of the sequence follow from this; for example, a(3*n+2) == 0 (mod 3), a(5*n+2) == a(5*n+3) (mod 5), a(7*n+1) == 0 (mod 7) and a(11*n+4) == 0 (mod 11). (End)

a(n) = (n*(n^2 + 3*n + 1)*a(n-1) - (n + 2))/(n^2 + n - 1), with a(0) = 2. - G. C. Greubel, Oct 07 2023

Maple

f:= rectoproc({a(n)=((4+3*n)*a(n-1)-(n+3)*(n-1)*a(n-2)+(n-1)*(n-2)*a(n-3))/2, a(0)=2, a(1)=7, a(2)=30}, a(n), remember):
map(f, [$0..40]); # Robert Israel, Oct 02 2016

Mathematica

(* First program *)
t[0, _] = 1; t[n_, 0] := t[n, 0] = t[n-1, 1];
t[n_, k_] := t[n, k] = t[n-1, k+1] + k*t[n, k-1];
a[n_] := t[2, n];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Aug 19 2022 *)
(* Second program *)
a[n_]:= a[n]= If[n==0, 2, (n*(n^2+3*n+1)*a[n-1] -(n+2))/(n^2+n-1)];
Table[a[n], {n, 0, 40}] (* G. C. Greubel, Oct 07 2023 *)

Prog

(Magma)
A144495:= func< n | (&+[Binomial(n, k)*(k+4)*Factorial(k+1) : k in [0..n]])/2 >;
[A144495(n): n in [0..40]]; // G. C. Greubel, Oct 07 2023
(SageMath)
def A144495(n): return sum(binomial(n, j)*factorial(j+1)*(j+4) for j in range(n+1))/2
[A144495(n) for n in range(41)] # G. C. Greubel, Oct 07 2023

Crossrefs

Cf. A000522, A001339, A038720, A144496, A144497, A144498.

Cf. A144499, A144500, A144501, A144502, A144503.

Sequence in context: A030848 A321735 A030975 * A154259 A066114 A088128

Adjacent sequences: A144492 A144493 A144494 * A144496 A144497 A144498

Keyword

nonn

Author

David Applegate and N. J. A. Sloane, Dec 13 2008

Status

approved