A300046 E.g.f. A(x) satisfies: A(x)^2 = 1 + Integral 2*A(4*x) dx.

1, 1, 3, 39, 2313, 579393, 589702779, 2411255004327, 39486660963182289, 2587442170418181942849, 678256560558087067898772147, 711196089994406513230761713249511, 2982964070560502237470559300237857712217, 50045793753333442365354429003593733008432339457, 3358515666122576624391239404741227000033885158827296171

Offset

0,3

Comments

Compare to: G(x)^2 = 1 + Integral 2*G(2*x) dx holds when G(x) = exp(x).

Links

Table of n, a(n) for n=0..14.

Formula

E.g.f.: A(x) = sqrt(G(2*x)) where G(x) is the e.g.f. of A300045.

a(n) = A300045(n+1) / 2^n.

Example

E.g.f. A(x) = 1 + x + 3*x^2/2! + 39*x^3/3! + 2313*x^4/4! + 579393*x^5/5! + 589702779*x^6/6! + 2411255004327*x^7/7! + 39486660963182289*x^8/8! + 2587442170418181942849*x^9/9! + ...
Related series.
A(x)^2 = 1 + 2*x + 8*x^2/2! + 96*x^3/3! + 4992*x^4/4! + 1184256*x^5/5! + 1186596864*x^6/6! + ... + 2*4^(n-1) * a(n-1)*x^n/n! + ...

Prog

(PARI) {a(n) = my(A=1+x); for(i=1, n, A = sqrt(1 + intformal(2*subst(A, x, 4*x) +x*O(x^n) ))); n!*polcoeff(A, n)}
for(n=0, 16, print1(a(n), ", "))

Crossrefs

Cf. A300045, A300047.

Sequence in context: A210921 A290516 A210925 * A267624 A342178 A342166

Adjacent sequences: A300043 A300044 A300045 * A300047 A300048 A300049

Keyword

nonn

Author

Paul D. Hanna, Feb 25 2018

Status

approved