A001517 Bessel polynomials y_n(x) (see A001498) evaluated at 2.
1, 3, 19, 193, 2721, 49171, 1084483, 28245729, 848456353, 28875761731, 1098127402131, 46150226651233, 2124008553358849, 106246577894593683, 5739439214861417731, 332993721039856822081, 20651350143685984386753
Offset: 0
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
Crossrefs
Programs
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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 *) -
PARI
a(n)=sum(k=0,n,(n+k)!/k!/(n-k)!)
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Sage
A001517 = lambda n: hypergeometric([-n, n+1], [], -1) [simplify(A001517(n)) for n in (0..16)] # Peter Luschny, Oct 17 2014
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
Extensions
More terms from Vladeta Jovovic, Apr 03 2000
Additional comments from Michael Somos, Jul 15 2002
Comments