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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;
text;
internal format)
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OFFSET
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0,2
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COMMENTS
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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
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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
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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
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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
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MATHEMATICA
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Table[(2k)! Hypergeometric1F1[-k, -2k, 1]/k!, {k, 0, 10}] (* Vladimir Reshetnikov, Feb 16 2011 *)
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PROG
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(PARI) a(n)=sum(k=0, n, (n+k)!/k!/(n-k)!)
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CROSSREFS
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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
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KEYWORD
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nonn,easy,nice,changed
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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More terms from Vladeta Jovovic, Apr 03 2000
Additional comments from Michael Somos, Jul 15, 2002.
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STATUS
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approved
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