This site is supported by donations to The OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A052143 E.g.f.: exp(x)/sqrt(1-4*x). 2
 1, 3, 17, 163, 2241, 39971, 874513, 22652547, 677742593, 22996109251, 872449527441, 36595485309923, 1681600030358977, 84005018253431523, 4532832802360066961, 262732854317051785411, 16280199853832658463233, 1073958487530496802770307 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 REFERENCES R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see page 191. LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..200 Vladimir Kruchinin, D. V. Kruchinin, Composita and their properties, arXiv:1103.2582 [math.CO], 2011-2013. FORMULA a(n) = n!*Sum_{k=0..n} A000984(k)/(n-k)!. - Vladimir Kruchinin, Sep 10 2010 a(n) = Sum_{k=0..n} binomial(n,k)*(2*k)!/k!. - Vladimir Kruchinin, Sep 10 2010 a(n) ~ sqrt(2)*4^n*n^n/exp(n-1/4). - Vaclav Kotesovec, Jun 27 2013 Conjecture: a(n) - (4*n-1)*a(n-1) + 4*(n-1)*a(n-2) = 0. - R. J. Mathar, Sep 27 2013 a(n) = U(1/2, n+3/2, 1/4)/2 where U denotes the Kummer U function. - Peter Luschny, Nov 26 2014 From Peter Bala, Nov 21 2017: (Start) a(n+k) = a(n) (mod k) for all n and k. It follows that the sequence a(n) taken modulo k is periodic with the exact period dividing k. For example, modulo 10 the sequence becomes 1, 3, 7, 3, 1, 1, 3, 7, 3, 1, ... with exact period 5. The e.g.f. A(x) = 1/sqrt(1 - 4*x)*exp(x) satisfies the differential equation (1 - 4*x)A' - (3 - 4*x)*A = 0 with A(0) = 1. Mathar's conjectured recurrence above follows from this. (End) MAPLE A052143 := n -> KummerU(1/2, n+3/2, 1/4)/2: seq(simplify(A052143(n)), n=0..17); # Peter Luschny, Dec 18 2017 MATHEMATICA CoefficientList[Series[E^x/Sqrt[1-4*x], {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jun 27 2013 *) PROG (Maxima) makelist(sum(binomial(n, k)*binomial(2*k, k)*(k)!, k, 0, n), n, 0, 12); /* Emanuele Munarini, Dec 17 2017 */ (PARI) x='x+O('x^99); Vec(serlaplace(exp(x)/sqrt(1-4*x))) \\ Altug Alkan, Dec 17 2017 CROSSREFS Cf. A000984. Sequence in context: A163884 A221410 A175607 * A268758 A069856 A214346 Adjacent sequences:  A052140 A052141 A052142 * A052144 A052145 A052146 KEYWORD nonn,easy AUTHOR N. J. A. Sloane, Jan 23 2000 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified November 18 07:39 EST 2018. Contains 317279 sequences. (Running on oeis4.)