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A082030 Expansion of e.g.f. exp(x)/(1-x)^3. 15
1, 4, 19, 106, 685, 5056, 42079, 390454, 4000441, 44881660, 547457611, 7215589954, 102211815589, 1548801969976, 25000879886935, 428332610385166, 7763306399014129, 148412806214119924, 2984692721713278211 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Binomial transform of A001710 (when preceded by 0).

From Peter Bala, 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} 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

Table of n, a(n) for n=0..18.

Roland Bacher, Counting Packings of Generic Subsets in Finite Groups, Electr. J. Combinatorics, 19 (2012), #P7. - From N. J. A. Sloane, Feb 06 2013

Eric Weisstein's World of Mathematics, Poisson-Charlier polynomial

FORMULA

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

a(n) = A001340(n)/2.

a(n) = Sum_{k=0..n} A046716(n, k)*3^(n-k). - Philippe Deléham, Jun 12 2004

a(n) = Sum_{k=0..n} binomial(n, k)*(k+2)!/2. - Philippe Deléham, Jun 19 2004

a(n) = Sum_{k=0..n} binomial(n,k)*(k+1)^(k+1)*(-k)^(n-k). - 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. - Paul D. Hanna, Sep 30 2011

Conjecture: a(n) + (-n-3)*a(n-1) + (n-1)*a(n-2) = 0. - R. J. Mathar, Dec 03 2012

G.f.: (1-x)/(2*x*Q(0)) - 1/2/x, where Q(k) = 1 - x - x*(k+2)/(1 - x*(k+1)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, Apr 22 2013

a(n) = hypergeometric([3, -n], [], -1). - Peter Luschny, Sep 20 2014

MAPLE

a := n -> hypergeom([3, -n], [], -1); seq(round(evalf(a(n), 100)), n=0..18); # Peter Luschny, Sep 20 2014

MATHEMATICA

a[n_] := a[n] = If[n == 0, 1, (n (n^2 + n + 1) a[n-1] + 1)/(n^2 - n + 1)];

a /@ Range[0, 18] (* Jean-François Alcover, Oct 16 2019 *)

PROG

(PARI) {a(n)=n!*polcoeff(exp(x+x*O(x^n))/(1-x)^3, n)} /* Paul D. Hanna, Sep 30 2011 */

(PARI) {a(n)=sum(k=0, n, binomial(n, k)*(k+2)!/2)} /* Paul D. Hanna, Sep 30 2011 */

(PARI) {a(n)=sum(k=0, n, binomial(n, k)*(k+1)^(k+1)*(-k)^(n-k))} /* Paul D. Hanna, Sep 30 2011 */

(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, Sep 30 2011 */

CROSSREFS

Cf. A082031, A000522, A001339, A095000, A095177.

Sequence in context: A186997 A062265 A088129 * A052751 A249934 A174992

Adjacent sequences:  A082027 A082028 A082029 * A082031 A082032 A082033

KEYWORD

easy,nonn

AUTHOR

Paul Barry, Apr 02 2003

STATUS

approved

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Last modified August 9 11:08 EDT 2020. Contains 336323 sequences. (Running on oeis4.)