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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

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