OFFSET
1,3
LINKS
Paul D. Hanna, Table of n, a(n) for n = 1..200
FORMULA
E.g.f. A(x) = Sum_{n>=1} a(n)*x^n/n! satisfies the following formulas, in which the constant of integration is zero where applicable.
(1) A'(x) = 1 + A(A(A(x))).
(2) A(x) = x + Integral A(A(A(x))) dx.
(3) A(x) = Series_Reversion( Integral 1/(1 + A(A(x))) dx ). - Paul D. Hanna, Jun 09 2026
From Seiichi Manyama, Jun 12 2026: (Start)
E.g.f. A(x) satisfies A'(x) = 1 + A^l(x), where A^l(x) denotes the l-th iterate of A, with A(0) = 0.
Let a(n,k,l) = n! * [x^n] A^k(x), where A^k(x) is the k-th iterate of A.
a(n,0,l) = 0^(n-1) and a(n,k,l) = a(n,k-1,l) + Sum_{j=1..n-1} binomial(n-1,j) * a(j,k+l-1,l) * a(n-j,k-1,l) for k > 0. (End)
EXAMPLE
E.g.f.: A(x) = x + x^2/2! + 3*x^3/3! + 18*x^4/4! + 171*x^5/5! + 2283*x^6/6! + ...
Related expansions.
The second iteration of A(x) begins
A(A(x)) = x + 2*x^2/2! + 9*x^3/3! + 69*x^4/4! + 777*x^5/5! + 11802*x^6/6! + 229047*x^7/7! + 5472600*x^8/8! + ...
where A(x) = Series_Reversion( Integral 1/(1 + A(A(x))) dx ).
The third iteration of A(x) begins
A(A(A(x))) = x + 3*x^2/2! + 18*x^3/3! + 171*x^4/4! + 2283*x^5/5! + ...
where A(x) = x + Integral A(A(A(x))) dx.
PROG
(PARI) {a(n) = my(A=x); for(k=1, n, A= truncate(A) + x*O(x^k);
A = intformal(1 + subst(A, x, subst(A, x, A))) +x*O(x^n) ); n!*polcoef(EGF=A, n)}
{upto(n) = a(n); Vec(serlaplace(EGF))}
upto(25) \\ program revised by Paul D. Hanna, Jun 09 2026
(PARI)
a_vector(n, k=1, l=3) = {
my(k_limit(row)=k+(n-row)*l, A=vector(n, row, vector(k_limit(row)+1)));
for(col=0, k_limit(1), A[1][col+1]=1);
for(row=2, n, A[row][1]=0);
for(row=2, n,
for(col=1, k_limit(row),
A[row][col+1]=A[row][col]+sum(j=1, row-1, binomial(row-1, j)*A[j][col+l]*A[row-j][col]);
);
);
vector(n, row, A[row][k+1])
}; \\ Seiichi Manyama, Jun 12 2026
CROSSREFS
KEYWORD
nonn,changed
AUTHOR
Paul D. Hanna, Jul 15 2011
STATUS
approved