A052143 E.g.f.: exp(x)/sqrt(1-4*x).
1, 3, 17, 163, 2241, 39971, 874513, 22652547, 677742593, 22996109251, 872449527441, 36595485309923, 1681600030358977, 84005018253431523, 4532832802360066961, 262732854317051785411, 16280199853832658463233, 1073958487530496802770307
Offset: 0
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.
Crossrefs
Cf. A000984.
Programs
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Maple
A052143 := n -> KummerU(1/2, n+3/2, 1/4)/2: seq(simplify(A052143(n)), n=0..17); # Peter Luschny, Dec 18 2017
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Mathematica
CoefficientList[Series[E^x/Sqrt[1-4*x], {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jun 27 2013 *) -
Maxima
makelist(sum(binomial(n,k)*binomial(2*k,k)*(k)!,k,0,n),n,0,12); /* Emanuele Munarini, Dec 17 2017 */
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PARI
x='x+O('x^99); Vec(serlaplace(exp(x)/sqrt(1-4*x))) \\ Altug Alkan, Dec 17 2017
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
D-finite with recurrence: 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 recurrence above follows from this. (End)