A143917 G.f. A(x) satisfies A(x) = 1/(1-x) + x^2*A(x)*A'(x).
1, 1, 2, 6, 25, 133, 851, 6313, 53061, 497493, 5144500, 58161126, 713789847, 9453038227, 134405493652, 2042529150110, 33045300698761, 567165849906233, 10294218618819268, 197022941365579804, 3966001076798967837, 83767346751954718361, 1852440991624711835677
Offset: 0
Keywords
Examples
G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 25*x^4 + 133*x^5 + 851*x^6 +... A'(x) = 1 + 4*x + 18*x^2 + 100*x^3 + 665*x^4 + 5106*x^5 +... A(x)*A'(x) = 1 + 5*x + 24*x^2 + 132*x^3 + 850*x^4 + 6312*x^5 +...
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..200
Programs
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Mathematica
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 *) -
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)} -
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
Formula
a(n) ~ c * n!, where c = 1.81857005675331400362707139219522893237... (see A238214). - Vaclav Kotesovec, Feb 20 2014
a(n) = 1 + Sum_{k=0..n-1} k * a(k) * a(n-1-k). - Seiichi Manyama, Jul 10 2025
a(n) = 1 + (n-1)/2 * Sum_{k=0..n-1} a(k) * a(n-1-k). - Seiichi Manyama, Jul 15 2025