0,2

Sum{k = 0..n} A094816(n,k)*x^k give A000522(n), A001339(n), A082030(n) for x = 1, 2, 3 respectively.

Recurrence relation: a(0) = 1, a(1) = 5, a(n) = (n+4)*a(n-1) - (n-1)*a(n-2) for n >= 2. Let p_3(n) = n^3+2*n-1 = n^(3)-3*n^(2)+3*n^(1)-1, where n^(k) denotes the rising factorial n*(n+1)*...*(n+k-1). The polynomial p_3(n) is an example of a Poisson-Charlier polynomial c_k(x;a) at k = 3, x = -n and a = -1.

The sequence b(n) := n!*p_3(n+1) = A001565(n) satisfies the same recurrence as a(n) but with the initial conditions b(0) = 2, b(1) = 11. This leads to the finite continued fraction expansion a(n)/b(n) = 1/(2+1/(5-1/(6-2/(7-...-(n-1)/(n+4))))).

Lim n -> infinity a(n)/b(n) = e/6 = 1/(2+1/(5-1/(6-2/(7-...-n/((n+5)-...))))).

a(n) = -b(n) * sum {k = 0..n} 1/(k!*p_3(k)*p_3(k+1)) - since the rhs satisfies the above recurrence with the same initial conditions. Hence e = -6 * sum {k = 0..inf} 1/(k!*p_3(k)*p_3(k+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), A082030 (r=2) and A095177 (r=4). (End)

Weisstein, Eric W., Poisson-Charlier polynomial

a(n) = Sum_{k = 0..n} A094816(n, k)*4^k . a(n) = Sum{k= 0..n} binomial(n, k)*(k+3)!/6.

Comments from Peter Bala (pbala@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.

Sequence in context: A078945 A113713 A062191 this_sequence A086672 A094710 A108453

Cf. A000522, A001339, A082030, A095177.

Adjacent sequences: A094997 A094998 A094999 this_sequence A095001 A095002 A095003

nonn

DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Jun 19 2004