|
| |
|
|
A082030
|
|
Expansion of exp(x)/(1-x)^3.
|
|
13
| |
|
|
1, 4, 19, 106, 685, 5056, 42079, 390454, 4000441, 44881660, 547457611, 7215589954, 102211815589, 1548801969976, 25000879886935, 428332610385166, 7763306399014129, 148412806214119924, 2984692721713278211
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,2
|
|
|
COMMENTS
| 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)
|
|
|
LINKS
| Weisstein, Eric W., Poisson-Charlier polynomial
|
|
|
FORMULA
| 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
a(n) = Sum_{k=0..n} binomial(n,k)*(k+1)^(k+1)*(-k)^(n-k). [From Paul D. Hanna, Sep 30 2011]
O.g.f.: Sum_{n>=0} (n+1)^(n+1)*x^n/(1+n*x)^(n+1) = Sum_{n>=0} a(n)*x^n. [From Paul D. Hanna, Sep 30 2011]
|
|
|
PROG
| (PARI) {a(n)=n!*polcoeff(exp(x+x*O(x^n))/(1-x)^3, n)}
(PARI) {a(n)=sum(k=0, n, binomial(n, k)*(k+2)!/2)}
(PARI) {a(n)=sum(k=0, n, binomial(n, k)*(k+1)^(k+1)*(-k)^(n-k))} /* Paul D. Hanna */
(PARI) {a(n)=polcoeff(sum(m=0, n, (m+1)^(m+1)*x^m/(1+m*x)^(m+1)+x*O(x^n)), n)} /* Paul D. Hanna */
|
|
|
CROSSREFS
| Cf. A082031.
Equals A001340(n)/2.
Cf. A000522, A001339, A095000, A095177.
Sequence in context: A186997 A062265 A088129 * A052751 A174992 A206227
Adjacent sequences: A082027 A082028 A082029 * A082031 A082032 A082033
|
|
|
KEYWORD
| easy,nonn
|
|
|
AUTHOR
| Paul Barry (pbarry(AT)wit.ie), Apr 02 2003
|
| |
|
|
