This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A294785 #20 Mar 15 2025 02:36:32
%S A294785 1,1,3,9,57,297,2187,15921,181233,1731249,20741139,241294329,
%T A294785 3524256297,49123306521,781173645723,12522002462433,247000850880993,
%U A294785 4516315005395169,92648539990208547,1886480713319540841,43524900326040674841,986331301183882645641,24094409085348757028523,596222660659090240456209,16242798073806940474325457,438933088683325211888103057,12586136448791084548892537907
%N A294785 E.g.f. A(x) satisfies: A(x) = A(x^2) * exp( Integral A(x^2) dx ).
%H A294785 Paul D. Hanna, <a href="/A294785/b294785.txt">Table of n, a(n) for n = 0..400</a>
%F A294785 E.g.f. satisfies:
%F A294785 (1) exp( Integral A(x^2) dx ) = 1 + Integral A(x) dx.
%F A294785 (2) A(x)/A(x^2) = 1 + Integral A(x) dx.
%F A294785 (3) A(x) = Product_{n>=0} B( x^(2^n) ) where B(x) = 1 + Integral A(x) dx.
%F A294785 (4) A'(x)/A(x) = A(x^2) + 2*x * A'(x^2)/A(x^2).
%F A294785 (5) A'(x)/A(x) = Sum_{n>=0} 2^n * x^(2^n-1) * A( x^(2^(n+1)) ).
%e A294785 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! +...
%e A294785 such that A(x) = A(x^2) * exp( Integral A(x^2) dx ).
%e A294785 Also,
%e A294785 A(x) = B(x) * B(x^2) * B(x^4) * B(x^8) * B(x^16) *...* B(x^(2^n)) *...
%e A294785 where B(x) = 1 + Integral A(x) dx.
%e A294785 Further,
%e A294785 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))) +...
%e A294785 RELATED SERIES.
%e A294785 E.g.f. A(x) as a series with reduced fractional coefficients begins:
%e A294785 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 +...
%e A294785 The logarithm of the e.g.f. begins:
%e A294785 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 +...
%e A294785 The logarithmic derivative of the e.g.f. begins:
%e A294785 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 +...
%e A294785 where A'(x)/A(x) = A(x^2) + 2*x * A'(x^2)/A(x^2).
%e A294785 The following series demonstrates an important property of the e.g.f.:
%e A294785 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)! +...
%e A294785 where A(x)/A(x^2) = 1 + Integral A(x) dx.
%o A294785 (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)}
%o A294785 for(n=0,30,print1(a(n),", "))
%Y A294785 Cf. A294638, A294784.
%K A294785 nonn
%O A294785 0,3
%A A294785 _Paul D. Hanna_, Nov 08 2017