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A001517 Bessel polynomials y_n(x) (see A001498) evaluated at 2.
(Formerly M3062 N1240)
15
1, 3, 19, 193, 2721, 49171, 1084483, 28245729, 848456353, 28875761731, 1098127402131, 46150226651233, 2124008553358849, 106246577894593683, 5739439214861417731, 332993721039856822081, 20651350143685984386753 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Numerators of successive convergents to e using continued fraction 1+2/(1+1/(6+1/(10+1/(14+1/(18+1/(22+1/26...)))))).

Number of ways to use the elements of {1,..,k}, n<=k<=2n, once each to form a collection of n lists, each having length 1 or 2. - Bob Proctor, Apr 18 2005, Jun 26 2006

REFERENCES

L. Euler, 1737.

J. W. L. Glaisher, Reports of British Assoc. Adv. Sci., 1871, pp. 16-18.

I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products, 6th ed., Section 0.126, p. 2.

D. H. Lehmer, Review of various tables by P. Pederson, Math. Comp., 2 (1946), 68-69.

J. Riordan, Combinatorial Identities, Wiley, 1968, p. 77.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

T. D. Noe, Table of n, a(n) for n=0..100

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 131

Index entries for related partition-counting sequences

Index entries for sequences related to Bessel functions or polynomials

FORMULA

a(n) = Sum_{k=0..n} (n+k)!/(k!*(n-k)!) = (e/pi)^(1/2) K_{n+1/2}(1/2).

a(n) = (4*n-2)*a(n-1) + a(n-2), n>=2.

a(n) = (1/n!)*Sum_{k=0..n} (-1)^(n+k)*binomial(n,k)*A000522(n+k). - Vladeta Jovovic, Sep 30 2006

E.g.f. exp(x*c(x)), where c(x)=(1-sqrt(1-4*x))/(2*x) (cf. A000108).  Vladimir Kruchinin, Aug 10 2010]

G.f.: 1/Q(0), where Q(k)= 1 - x - 2*x*(k+1)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 17 2013

MATHEMATICA

Table[(2k)! Hypergeometric1F1[-k, -2k, 1]/k!, {k, 0, 10}] (* Vladimir Reshetnikov, Feb 16 2011 *)

PROG

(PARI) a(n)=sum(k=0, n, (n+k)!/k!/(n-k)!)

CROSSREFS

Essentially the same as A080893.

a(n) = A099022(n)/n!.

Partial sums: A105747.

Replace "lists" by "sets" in comment: A001515.

Cf. A001515, A001518, A002119, A053556, A053557.

Sequence in context: A155805 A218261 * A080893 A028854 A222865 A108292

Adjacent sequences:  A001514 A001515 A001516 * A001518 A001519 A001520

KEYWORD

nonn,easy,nice,changed

AUTHOR

N. J. A. Sloane.

EXTENSIONS

More terms from Vladeta Jovovic, Apr 03 2000

Additional comments from Michael Somos, Jul 15, 2002.

STATUS

approved

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Last modified May 21 12:55 EDT 2013. Contains 225488 sequences.