cp's OEIS Frontend

A143917 G.f. A(x) satisfies A(x) = 1/(1-x) + x^2*A(x)*A'(x).

%I A143917 #31 Jul 15 2025 08:29:31
%S A143917 1,1,2,6,25,133,851,6313,53061,497493,5144500,58161126,713789847,
%T A143917 9453038227,134405493652,2042529150110,33045300698761,567165849906233,
%U A143917 10294218618819268,197022941365579804,3966001076798967837,83767346751954718361,1852440991624711835677
%N A143917 G.f. A(x) satisfies A(x) = 1/(1-x) + x^2*A(x)*A'(x).
%H A143917 Vincenzo Librandi, <a href="/A143917/b143917.txt">Table of n, a(n) for n = 0..200</a>
%F A143917 a(n) ~ c * n!, where c = 1.81857005675331400362707139219522893237... (see A238214). - _Vaclav Kotesovec_, Feb 20 2014
%F A143917 a(n) = 1 + Sum_{k=0..n-1} k * a(k) * a(n-1-k). - _Seiichi Manyama_, Jul 10 2025
%F A143917 a(n) = 1 + (n-1)/2 * Sum_{k=0..n-1} a(k) * a(n-1-k). - _Seiichi Manyama_, Jul 15 2025
%e A143917 G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 25*x^4 + 133*x^5 + 851*x^6 +...
%e A143917 A'(x) = 1 + 4*x + 18*x^2 + 100*x^3 + 665*x^4 + 5106*x^5 +...
%e A143917 A(x)*A'(x) = 1 + 5*x + 24*x^2 + 132*x^3 + 850*x^4 + 6312*x^5 +...
%t A143917 Clear[a]; a[0] = 1; a[n_]/; n>=1 := a[n] = 1 + Sum[(k - 1) a[k - 1] a[n - k], {k, n}]; Table[a[n], {n,0, 16}] (* _David Callan_, Jun 24 2013 *)
%o A143917 (PARI) {a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=1/(1-x+x*O(x^n))+x^2*A*deriv(A)); polcoeff(A, n)}
%o A143917 (PARI) a_vector(n) = my(v=vector(n+1)); for(i=0, n, v[i+1]=1+sum(j=0, i-1, j*v[j+1]*v[i-j])); v; \\ _Seiichi Manyama_, Jul 10 2025
%Y A143917 Cf. A143916 (variant), A238214.
%K A143917 nonn
%O A143917 0,3
%A A143917 _Paul D. Hanna_, Sep 05 2008