A300045 E.g.f. A(x) satisfies: A(x) = 1 + Integral A(4*x)^(1/2) dx.

1, 1, 2, 12, 312, 37008, 18540576, 37740977856, 308640640553856, 10108585206574665984, 1324770391254109154738688, 694534718011481157528342678528, 1456529592308544539096599988734998528, 12218220833015817164679410893774265189240832, 409975142427307559856983482397439860805077724831744, 55025920673752295414026066407280263168555174442226420465664

Offset

0,3

Comments

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

Links

Table of n, a(n) for n=0..15.

Example

E.g.f.: A(x) = 1 + x + 2*x^2/2! + 12*x^3/3! + 312*x^4/4! + 37008*x^5/5! + 18540576*x^6/6! + 37740977856*x^7/7! + 308640640553856*x^8/8! + 10108585206574665984*x^9/9! + ...
Related series.
A(4*x)^(1/2) = 1 + 2*x + 12*x^2/2! + 312*x^3/3! + 37008*x^4/4! + 18540576*x^5/5! + 37740977856*x^6/6! + ... + a(n+1)*x^n/n! + ...
A(2*x)^(1/2) = 1 + x + 3*x^2/2! + 39*x^3/3! + 2313*x^4/4! + 579393*x^5/5! + 589702779*x^6/6! + ... + A300046(n)*x^n/n! + ...
log(A(x)) = x + x^2/2! + 8*x^3/3! + 270*x^4/4! + 35472*x^5/5! + 18318288*x^6/6! + ... + A300047(n)*x^n/n! + ...

Prog

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

Crossrefs

Cf. A300046, A300047.

Sequence in context: A122767 A260321 A094047 * A091472 A156518 A012727

Adjacent sequences: A300042 A300043 A300044 * A300046 A300047 A300048

Keyword

nonn

Author

Paul D. Hanna, Feb 25 2018

Status

approved