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A001517
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Bessel polynomials y_n(x) (see A001498) evaluated at 2.
(Formerly M3062 N1240)
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15
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1, 3, 19, 193, 2721, 49171, 1084483, 28245729, 848456353, 28875761731, 1098127402131, 46150226651233, 2124008553358849, 106246577894593683, 5739439214861417731, 332993721039856822081, 20651350143685984386753
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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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
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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).
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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
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FORMULA
| a(n) = Sum_{k=0..n} (n+k)!/(k!*(n-k)!) = (e/pi)^(1/2) K_{n+1/2}(1/2).
a(n) = (4n-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 (vladeta(AT)eunet.rs), Sep 30 2006
E.g.f. exp(xCatalan(x)), Catalan(x)=(1-sqrt(1-4x))/2x, (A000108) [From Kruchinin Vladimir (kru(AT)ie.tusur.ru), Aug 10 2010]
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MATHEMATICA
| Table[(2k)! Hypergeometric1F1[-k, -2k, 1]/k!, {k, 0, 10}] (* Vladimir Reshetnikov (v.reshetnikov(AT)gmail.com), Feb 16 2011 *)
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PROG
| (PARI) a(n)=sum(k=0, n, (n+k)!/k!/(n-k)!)
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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: A101481 A155805 * A080893 A028854 A108292 A053554
Adjacent sequences: A001514 A001515 A001516 * A001518 A001519 A001520
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KEYWORD
| nonn,easy,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from Vladeta Jovovic (vladeta(AT)eunet.rs), Apr 03 2000
Additional comments from Michael Somos, Jul 15, 2002.
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