0,2

Binomial transform of A001710 (when preceded by 0)

Comments from Peter Bala (pbala(AT)toucansurf.com), Jul 10 2008 (Start): a(n) is a difference divisibility sequence, that is, the difference a(n) - a(m) is divisible by n - m for all n and m (provided n is not equal to m). See A000522 for further properties of difference divisibility sequences.

Recurrence relation: a(0) = 1, a(1) = 4, a(n) = (n+3)*a(n-1) - (n-1)*a(n-2) for n >=2. The sequence b(n) :=n!*(n^2+n+1) = A001564(n) satisfies the same recurrence with the initial conditions b(0) = 1, b(1) = 3. This leads to the finite continued fraction expansion a(n)/b(n) = 1/(1-1/(4-1/(5-2/(6-...-(n-1)/(n+3))))).

Lim n -> infinity a(n)/b(n) = e/2 = 1/(1-1/(4-1/(5-2/(6-...-n/((n+4)-...))))).

a(n) = n!*(n^2+n+1)*sum {k = 0..n} 1/(k!*(k^4+k^2+1)) - since the rhs satisfies the above recurrence with the same initial conditions. Hence e = 2*sum {k = 0..inf} 1/(k!*(k^4+k^2+1)).

For sequences satisfying the more general recurrence a(n) = (n+1+r)*a(n-1) - (n-1)*a(n-2), which yield series acceleration formulas for e/r! that involve the Poisson-Charlier polynomials c_r(-n;-1), refer to A000522 (r = 0), A001339 (r=1), A095000 (r=3) and A095177 (r=4). (End)

Weisstein, Eric W., Poisson-Charlier polynomial

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

a(n) = Sum_{k = 0..n} A046716(n, k)*3^(n-k) . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Jun 12 2004

a(n) = Sum_{k = 0..n} binomial(n, k)*(k+2)!/2. - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Jun 19 2004

Cf. A082031.

Equals A001340(n)/2.

Cf. A000522, A001339, A095000, A095177.

Sequence in context: A110531 A062265 A088129 this_sequence A052751 A091643 A117397

Adjacent sequences: A082027 A082028 A082029 this_sequence A082031 A082032 A082033

easy,nonn

Paul Barry (pbarry(AT)wit.ie), Apr 02 2003