A001517 Bessel polynomials y_n(x) (see A001498) evaluated at 2. (Formerly M3062 N1240)

1, 3, 19, 193, 2721, 49171, 1084483, 28245729, 848456353, 28875761731, 1098127402131, 46150226651233, 2124008553358849, 106246577894593683, 5739439214861417731, 332993721039856822081, 20651350143685984386753

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.

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

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

Robert Israel, Table of n, a(n) for n = 0..334 (first 101 terms from T. D. Noe)

P. Bala, A note on the Catalan transform of a sequence.

J. W. L. Glaisher, On Lambert's proof of the irrationality of Pi and on the irrationality of certain other quantities, Reports of British Assoc. Adv. Sci., 1871, pp. 16-18.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 131.

D. H. Lehmer, Arithmetical periodicities of Bessel functions, Annals of Mathematics, 33 (1932): 143-150. The sequence is on page 149.

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

W. Mlotkowski, A. Romanowicz, A family of sequences of binomial type, Probability and Mathematical Statistics, Vol. 33, Fasc. 2 (2013), pp. 401-408.

R. A. Proctor, Let's Expand Rota's Twelvefold Way for Counting Partitions!, arXiv:math/0606404 [math.CO], 2006-2007.

J. Riordan, Letter to N. J. A. Sloane, Jul. 1968.

J. Riordan, Letter, Jul 06 1978.

N. J. A. Sloane, Letter to J. Riordan, Nov. 1970.

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).

D-finite with recurrence 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. (for offset 1): 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

a(n) = (1/n!)*Integral_{x>=0} (x*(1 + x))^n*exp(-x) dx. Expansion of exp(x) in powers of y = x*(1 - x): exp(x) = 1 + y + 3*y^2/2! + 19*y^3/3! + 193*y^4/4! + 2721*y^5/5! + .... - Peter Bala, Dec 15 2013

a(n) = exp(1/2) / sqrt(Pi) * BesselK(n+1/2, 1/2). - Vaclav Kotesovec, Mar 15 2014

a(n) ~ 2^(2*n+1/2) * n^n / exp(n-1/2). - Vaclav Kotesovec, Mar 15 2014

a(n) = hypergeom([-n, n+1], [], -1). - Peter Luschny, Oct 17 2014

From G. C. Greubel, Aug 16 2017: (Start)

a(n) = (1/2)_{n} * 4^n * hypergeometric1f1(-n; -2*n; 1).

G.f.: (1/(1-t))*hypergeometric2f0(1, 1/2; -; 4*t/(1-t)^2). (End)

a(n) = Sum_{k=0..n} binomial(n,k)*binomial(n+k,k)*k!. - Ilya Gutkovskiy, Nov 24 2017

a(n) = KummerU(-n, -2*n, 1). - Peter Luschny, May 10 2022

Maple

A:= gfun:-rectoproc({a(n) = (4*n-2)*a(n-1) + a(n-2), a(0)=1, a(1)=3}, a(n), remember):
map(A, [$0..20]); # Robert Israel, Jul 22 2015
f:=proc(n) option remember; if n = 0 then 1 elif n=1 then 3 else f(n-2)+(4*n-2)*f(n-1); fi; end;
[seq(f(n), n=0..20)]; # N. J. A. Sloane, May 09 2016
seq(simplify(KummerU(-n, -2*n, 1)), n = 0..16); # Peter Luschny, May 10 2022

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)!)
(Sage)
A001517 = lambda n: hypergeometric([-n, n+1], [], -1)
[simplify(A001517(n)) for n in (0..16)] # Peter Luschny, Oct 17 2014

Crossrefs

Essentially the same as A080893.

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

Partial sums: A105747.

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

Cf. A001515, A001518, A002119, A053556, A053557, A243593.

Sequence in context: A101481 A155805 A218261 * A080893 A028854 A222865

Adjacent sequences: A001514 A001515 A001516 * A001518 A001519 A001520

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Extensions

More terms from Vladeta Jovovic, Apr 03 2000

Additional comments from Michael Somos, Jul 15 2002

Status

approved