login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A118930 E.g.f.: A(x) = exp( Sum_{n>=0} x^(2^n)/2^(2^n-1) ). 7
1, 1, 2, 4, 13, 41, 166, 652, 3494, 18118, 114076, 681176, 5016892, 35377564, 288204008, 2232198256, 21124254181, 191779964597, 2011347229114, 19840403629108, 231266808172181, 2553719667653281, 31743603728993542 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Equals invariant column vector V that satisfies matrix product A100861*V = V, where Bessel numbers A100861(n,k) = n!/[k!(n-2k)!*2^k] give the number of k-matchings of the complete graph K(n).

Equals Lim_{n->inf.} A144299^n, if A144299 is considered an infinite lower triangular matrix. - Gary W. Adamson, Dec 08 2008

LINKS

Table of n, a(n) for n=0..22.

FORMULA

a(n) = Sum_{k=0..[n/2]} n!/[k!*(n-2*k)!*2^k] * a(k), with a(0)=1. a(n) = Sum_{k=0..[n/2]} A100861(n,k)*a(k), with a(0)=1.

EXAMPLE

E.g.f. A(x) = exp( x + x^2/2 + x^4/2^3 + x^8/2^7 + x^16/2^15 +...)

= 1 + 1*x + 2*x^2/2! + 4*x^3/3! + 13*x^4/4! + 41*x^5/5!+ 166*x^6/6!+...

Using coefficients A100861(n,k) = n!/[k!(n-2k)!*2^k]:

a(5) = 1*a(0) +10*a(1) +15*a(2) = 1*1 +10*1 +15*2 = 41.

a(6) = 1*a(0) +15*a(1) +45*a(2) +15*a(3) = 1*1 +15*1 +45*2 +15*4 = 166.

MAPLE

A118930 := proc(n)

    option remember;

    if n<= 1 then

        1 ;

    else

        n!*add(procname(k)/k!/(n-2*k)!/2^k, k=0..n/2) ;

    end if;

end proc;

seq(A118930(n), n=0..10) ; # R. J. Mathar, Aug 19 2014

MATHEMATICA

a[n_] := a[n] = If[n==0, 1, Sum[Binomial[n, 2k] (2k-1)!! a[k], {k, 0, n/2}]];

a /@ Range[0, 22] (* Jean-Fran├žois Alcover, Mar 26 2020 *)

PROG

(PARI) {a(n)=if(n==0, 1, sum(k=0, n\2, n!/(k!*(n-2*k)!*2^k)*a(k)))}

for(n=0, 30, print1(a(n), ", "))

(PARI) /* Defined by E.G.F.: */

{a(n)=n!*polcoeff( exp(sum(k=0, #binary(n), x^(2^k)/2^(2^k-1))+x*O(x^n)), n, x)}

for(n=0, 30, print1(a(n), ", "))

CROSSREFS

Cf. A100861; variants: A118932, A118935.

Equals row sums of triangle A152685. - Gary W. Adamson, Dec 10 2008

Cf. A144299. - Gary W. Adamson, Dec 08 2008

Sequence in context: A148256 A163136 A325578 * A087214 A259239 A243107

Adjacent sequences:  A118927 A118928 A118929 * A118931 A118932 A118933

KEYWORD

nonn,changed

AUTHOR

Paul D. Hanna, May 06 2006

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
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified April 7 01:11 EDT 2020. Contains 333291 sequences. (Running on oeis4.)