A088221 Coefficient of x^n in g.f.^n is A000698(n+1).

1, 2, 3, 10, 63, 558, 6226, 82836, 1272555, 22103638, 427715118, 9118752300, 212335628550, 5362040637900, 145970732893284, 4261945511044520, 132868133756374707, 4405535689300995942, 154819142574597555670

Offset: 0

Author

Michael Somos, Sep 24 2003

Keywords

• nonn

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.

Crossrefs

Cf. A000698, A000699.

Programs

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 *)

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

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