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

0,2

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

Cf. A000698, A000699.

Sequence in context: A066526 A093856 A173097 * A206296 A124923 A291935

Adjacent sequences: A088218 A088219 A088220 * A088222 A088223 A088224

Keyword

nonn

Author

Michael Somos, Sep 24 2003

Status

approved