A230318 E.g.f. satisfies: A(x) = Sum_{n>=0} x^n/n! * Product_{k=1..n} A(k*x)^(1/k).

1, 1, 3, 16, 137, 1806, 36307, 1120099, 53543737, 3987172576, 462225323291, 83073483132399, 23029319036705797, 9800155391415000808, 6374883018317726171183, 6314677980863430818696221, 9492558648757349054617246833, 21588387865629652184260447567056

Offset

0,3

Links

Table of n, a(n) for n=0..17.

Example

G.f.: A(x) = 1 + x + 3*x^2/2! + 16*x^3/3! + 137*x^4/4! + 1806*x^5/5! + ...
where
A(x) = 1 + x*A(x) + x^2*A(x)*A(2*x)^(1/2)/2! + x^3*A(x)*A(2*x)^(1/2)*A(3*x)^(1/3)/3! + x^4*A(x)*A(2*x)^(1/2)*A(3*x)^(1/3)*A(4*x)^(1/4)/4! + ...
RELATED EXPANSIONS.
A(2*x)^(1/2) = 1 + x + 5*x^2/2! + 49*x^3/3! + 825*x^4/4! + 22321*x^5/5! + ...
A(3*x)^(1/3) = 1 + x + 7*x^2/2! + 100*x^3/3! + 2521*x^4/4! + 103606*x^5/5! + ...
A(4*x)^(1/4) = 1 + x + 9*x^2/2! + 169*x^3/3! + 5681*x^4/4! + 313521*x^5/5! + ...
A(5*x)^(1/5) = 1 + x + 11*x^2/2! + 256*x^3/3! + 10761*x^4/4! + 745726*x^5/5! + ...
A(6*x)^(1/6) = 1 + x + 13*x^2/2! + 361*x^3/3! + 18217*x^4/4! + 1519681*x^5/5! + ...
A(7*x)^(1/7) = 1 + x + 15*x^2/2! + 484*x^3/3! + 28505*x^4/4! + 2780646*x^5/5! + ...
A(8*x)^(1/8) = 1 + x + 17*x^2/2! + 625*x^3/3! + 42081*x^4/4! + 4699681*x^5/5! + ...
A(9*x)^(1/9) = 1 + x + 19*x^2/2! + 784*x^3/3! + 59401*x^4/4! + 7473646*x^5/5! + ...
A(10*x)^(1/10) = 1 + x + 21*x^2/2! + 961*x^3/3! + 80921*x^4/4! + 11325201*x^5/5! + ...

Prog

(PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+sum(m=1, n, x^m/m!*prod(k=1, m, subst(A, x, k*x+x*O(x^n))^(1/k)))); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))

Crossrefs

Cf. A230320, A230317.

Sequence in context: A368293 A345349 A230320 * A006057 A305472 A002719

Adjacent sequences: A230315 A230316 A230317 * A230319 A230320 A230321

Keyword

nonn

Author

Paul D. Hanna, Oct 15 2013

Status

approved