OFFSET
0,2
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..250
Ali Assem Mahmoud, On the Asymptotics of Connected Chord Diagrams, University of Waterloo (Ontario, Canada 2019).
Ali Assem Mahmoud and Karen Yeats, Connected Chord Diagrams and the Combinatorics of Asymptotic Expansions, arXiv:2010.06550 [math.CO], 2020.
FORMULA
a(n) = Sum_{j=1..n-1} (4*j-1)*A000699(j)*A000699(n-j), with a(0)=1, a(1)=2. - G. C. Greubel, Feb 08 2020
MAPLE
c:= proc(n) option remember;
if n=1 then 1
else (n-1)*add( c(j)*c(n-j), j=1..n-1)
fi; end:
a:= proc(n) option remember;
if n<2 then n+1
else add( (4*j-1)*c(j)*c(n-j), j=1..n-1)
fi; end;
seq(a(n), n=0..20); # G. C. Greubel, Feb 08 2020
MATHEMATICA
c[n_]:= c[n]= If[n==1, 1, (n-1)*Sum[c[j]*c[n-j], {j, n-1}]];
a[n_]:= If[n<2, n+1, Sum[(4*j-1)*c[j]*c[n-j], {j, n-1}]];
Table[a[n], {n, 0, 20}] (* G. C. Greubel, Feb 08 2020 *)
PROG
(Sage)
@CachedFunction
def c(n):
if (n==1): return 1
else: return (n-1)*sum( c(j)*c(n-j) for j in (1..n-1) )
def a(n):
if (n<2): return n+1
else: return sum( (4*j-1)*c(j)*c(n-j) for j in (1..n-1) )
[a(n) for n in (0..20)] # G. C. Greubel, Feb 08 2020
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael Somos, Sep 24 2003
STATUS
approved