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

%I A088221 #11 Oct 13 2020 23:49:15
%S A088221 1,2,3,10,63,558,6226,82836,1272555,22103638,427715118,9118752300,
%T A088221 212335628550,5362040637900,145970732893284,4261945511044520,
%U A088221 132868133756374707,4405535689300995942,154819142574597555670
%N A088221 Coefficient of x^n in g.f.^n is A000698(n+1).
%H A088221 G. C. Greubel, <a href="/A088221/b088221.txt">Table of n, a(n) for n = 0..250</a>
%H A088221 Ali Assem Mahmoud, <a href="https://uwaterloo.ca/scholar/a39mahmo/publications/aymptotics-connected-chord-diagrams">On the Asymptotics of Connected Chord Diagrams</a>, University of Waterloo (Ontario, Canada 2019).
%H A088221 Ali Assem Mahmoud and Karen Yeats, <a href="https://arxiv.org/abs/2010.06550">Connected Chord Diagrams and the Combinatorics of Asymptotic Expansions</a>, arXiv:2010.06550 [math.CO], 2020.
%F A088221 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
%p A088221 c:= proc(n) option remember;
%p A088221       if n=1 then 1
%p A088221     else (n-1)*add( c(j)*c(n-j), j=1..n-1)
%p A088221       fi; end:
%p A088221 a:= proc(n) option remember;
%p A088221         if n<2 then n+1
%p A088221       else add( (4*j-1)*c(j)*c(n-j), j=1..n-1)
%p A088221         fi; end;
%p A088221 seq(a(n), n=0..20); # _G. C. Greubel_, Feb 08 2020
%t A088221 c[n_]:= c[n]= If[n==1, 1, (n-1)*Sum[c[j]*c[n-j], {j,n-1}]];
%t A088221 a[n_]:= If[n<2, n+1, Sum[(4*j-1)*c[j]*c[n-j], {j,n-1}]];
%t A088221 Table[a[n], {n, 0, 20}] (* _G. C. Greubel_, Feb 08 2020 *)
%o A088221 (Sage)
%o A088221 @CachedFunction
%o A088221 def c(n):
%o A088221     if (n==1): return 1
%o A088221     else: return (n-1)*sum( c(j)*c(n-j) for j in (1..n-1) )
%o A088221 def a(n):
%o A088221     if (n<2): return n+1
%o A088221     else: return sum( (4*j-1)*c(j)*c(n-j) for j in (1..n-1) )
%o A088221 [a(n) for n in (0..20)] # _G. C. Greubel_, Feb 08 2020
%Y A088221 Cf. A000698, A000699.
%K A088221 nonn
%O A088221 0,2
%A A088221 _Michael Somos_, Sep 24 2003