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A144495
Row 2 of array in A144502.
5
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
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
KEYWORD
nonn
AUTHOR
STATUS
approved