OFFSET
0,3
FORMULA
a(n) = Sum_{k=1..n} k(k+1)/2 * C(n-1,k-1)*a(k-1)*a(n-k) for n>0, with a(0)=1.
E.g.f. satisifies: A(x) = exp( d/dx x^2*A(x)/2 ). - Paul D. Hanna, Dec 17 2017
EXAMPLE
E.g.f: A(x) = 1 + x + 4*x^2/2! + 34*x^3/3! + 482*x^4/4! + 10056*x^5/5! +...
log(A(x)) = x + 3*1*x^2/2! + 6*4*x^3/3! + 10*34*x^4/4! + 15*482*x^5/5! +...
such that log(A(x)) = x*A(x) + x^2*A'(x)/2 = d/dx x^2*A(x)/2.
PROG
(PARI) {a(n) = if(n==0, 1, n!*polcoeff(exp(sum(k=1, n, k*(k+1)/2*a(k-1)*x^k/k!)+x*O(x^n)), n))}
for(n=0, 25, print1(a(n), ", "))
(PARI) {a(n) = if(n==0, 1, sum(k=1, n, k*(k+1)/2*binomial(n-1, k-1)*a(k-1)*a(n-k)))}
for(n=0, 25, print1(a(n), ", "))
(PARI) {a(n) = my(A=1); for(i=1, n, A = exp(deriv(x^2*A/2 +x^2*O(x^n)))); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", ")) \\ Paul D. Hanna, Dec 17 2017
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Feb 08 2009
STATUS
approved