A294785 E.g.f. A(x) satisfies: A(x) = A(x^2) * exp( Integral A(x^2) dx ).

1, 1, 3, 9, 57, 297, 2187, 15921, 181233, 1731249, 20741139, 241294329, 3524256297, 49123306521, 781173645723, 12522002462433, 247000850880993, 4516315005395169, 92648539990208547, 1886480713319540841, 43524900326040674841, 986331301183882645641, 24094409085348757028523, 596222660659090240456209, 16242798073806940474325457, 438933088683325211888103057, 12586136448791084548892537907

Offset

0,3

Links

Paul D. Hanna, Table of n, a(n) for n = 0..400

Formula

E.g.f. satisfies:

(1) exp( Integral A(x^2) dx ) = 1 + Integral A(x) dx.

(2) A(x)/A(x^2) = 1 + Integral A(x) dx.

(3) A(x) = Product_{n>=0} B( x^(2^n) ) where B(x) = 1 + Integral A(x) dx.

(4) A'(x)/A(x) = A(x^2) + 2*x * A'(x^2)/A(x^2).

(5) A'(x)/A(x) = Sum_{n>=1} 2^n * x^(2^n-1) * A( x^(2^(n+1) ).

Example

E.g.f.: A(x) = 1 + x + 3*x^2/2! + 9*x^3/3! + 57*x^4/4! + 297*x^5/5! + 2187*x^6/6! + 15921*x^7/7! + 181233*x^8/8! + 1731249*x^9/9! + 20741139*x^10/10! + 241294329*x^11/11! + 3524256297*x^12/12! + 49123306521*x^13/13! + 781173645723*x^14/14! + 12522002462433*x^15/15! + 247000850880993*x^16/16! +...
such that A(x) = A(x^2) * exp( Integral A(x^2) dx ).
Also,
A(x) = B(x) * B(x^2) * B(x^4) * B(x^8) * B(x^16) *...* B(x^(2^n)) *...
where B(x) = 1 + Integral A(x) dx.
Further,
A'(x)/A(x) = A(x^2) + 2*x*A(x^4) + 4*x^3*A(x^8) + 8*x^7*A(x^16) + 16*x^15*A(x^32) + 32*x^31*A(x^64) +...+ 2^n * x^(2^n-1) * A(x^(2^(n+1))) +...
RELATED SERIES.
E.g.f. A(x) as a series with reduced fractional coefficients begins:
A(x) = 1 + x + 3/2*x^2 + 3/2*x^3 + 19/8*x^4 + 99/40*x^5 + 243/80*x^6 + 1769/560*x^7 + 20137/4480*x^8 + 192361/40320*x^9 + 2304571/403200*x^10 + 8936827/1478400*x^11 + 43509337/5913600*x^12 + 1819381723/230630400*x^13 + 3214706361/358758400*x^14 + 51530874331/5381376000*x^15 + 277217565523/23482368000*x^16 +...
The logarithm of the e.g.f. begins:
log(A(x)) = x + x^2 + 1/3*x^3 + x^4 + 3/10*x^5 + 1/3*x^6 + 3/14*x^7 + x^8 + 19/72*x^9 + 3/10*x^10 + 9/40*x^11 + 1/3*x^12 + 243/1040*x^13 + 3/14*x^14 + 1769/8400*x^15 + x^16 + 20137/76160*x^17 + 19/72*x^18 + 192361/766080*x^19 + 3/10*x^20 + 2304571/8467200*x^21 + 9/40*x^22 + 8936827/34003200*x^23 + 1/3*x^24 + 43509337/147840000*x^25 + 243/1040*x^26 + 1819381723/6227020800*x^27 + 3/14*x^28 + 3214706361/10403993600*x^29 + 1769/8400*x^30 + 51530874331/166822656000*x^31 + x^32 +...
The logarithmic derivative of the e.g.f. begins:
A'(x)/A(x) = 1 + 2*x + x^2 + 4*x^3 + 3/2*x^4 + 2*x^5 + 3/2*x^6 + 8*x^7 + 19/8*x^8 + 3*x^9 + 99/40*x^10 + 4*x^11 + 243/80*x^12 + 3*x^13 + 1769/560*x^14 + 16*x^15 + 20137/4480*x^16 +...
where A'(x)/A(x) = A(x^2) + 2*x * A'(x^2)/A(x^2).
The following series demonstrates an important property of the e.g.f.:
A(x)/A(x^2) = 1 + x + x^2/2! + 3*x^3/3! + 9*x^4/4! + 57*x^5/5! + 297*x^6/6! + 2187*x^7/7! + 15921*x^8/8! + 181233*x^9/9! +...+ a(n)*x^(n+1)/(n+1)! +...
where A(x)/A(x^2) = 1 + Integral A(x) dx.

Prog

(PARI) {a(n) = my(A=1); for(i=1, #binary(n+1), A = subst(A, x, x^2) * exp( intformal( subst(A, x, x^2) +x*O(x^n))) ); n!*polcoeff(H=A, n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A294638, A294784.

Sequence in context: A075979 A128681 A292333 * A040175 A192252 A363011

Adjacent sequences: A294782 A294783 A294784 * A294786 A294787 A294788

Keyword

nonn

Author

Paul D. Hanna, Nov 08 2017

Status

approved