The OEIS is supported by the many generous donors to the OEIS Foundation.
%I #18 Nov 16 2017 22:36:51
%S 1,1,1,3,9,33,153,963,6129,47457,393489,3689379,36673209,410924097,
%T 4810169961,64694478627,878318278497,13230037503297,203967546446241,
%U 3494178651687363,60117798742663401,1137159539308348641,21683284489630748601,452680959717183978243,9454328250188008785489,214087305044257976127393,4862802200825123466537393,119970186740330465448543843,2944202974922987534742898329
%N E.g.f. satisfies: A'(x) = A(x) * A(x^2).
%H Paul D. Hanna, <a href="/A294638/b294638.txt">Table of n, a(n) for n = 0..520</a>
%F E.g.f. A(x) = Sum_{n>=0} a(n) * x^n/n! satisfies:
%F (1) A(x) = exp( Integral A(x^2) dx ).
%F (2) A(x) = 1/A(-x).
%F (3) A(x) = exp( Sum_{n>=0} a(n) * x^(2*n+1) / ((2*n+1)*n!) ) .
%F (4) A(x) = exp( Sum_{n>=0} (2*n)!/n! * a(n) * x^(2*n+1)/(2*n+1)! ).
%e E.g.f.: A(x) = 1 + x + x^2/2! + 3*x^3/3! + 9*x^4/4! + 33*x^5/5! + 153*x^6/6! + 963*x^7/7! + 6129*x^8/8! + 47457*x^9/9! + 393489*x^10/10! + 3689379*x^11/11! + 36673209*x^12/12! + 410924097*x^13/13! + 4810169961*x^14/14! + 64694478627*x^15/15! + 878318278497*x^16/16! + 13230037503297*x^17/17! + 203967546446241*x^18/18! + 3494178651687363*x^19/19! + ...
%e such that A'(x) = A(x) * A(x^2).
%e Also, A(x) = exp( Integral A(x^2) dx ).
%e RELATED SERIES.
%e The logarithm of the e.g.f. is an odd function that begins:
%e log(A(x)) = x + x^3/3 + x^5/(5*2!) + 3*x^7/(7*3!) + 9*x^9/(9*4!) + 33*x^11/(11*5!) + 153*x^13/(13*6!) + 963*x^15/(15*7!) + 6129*x^17/(17*8!) + 47457*x^19/(19*9!) + 393489*x^21/(21*10!) +...+ a(n) * x^(2*n+1)/((2*n+1)*n!) +...
%e which equals Integral A(x^2) dx.
%e Explicitly,
%e log(A(x)) = x + 2*x^3/3! + 12*x^5/5! + 360*x^7/7! + 15120*x^9/9! + 997920*x^11/11! + 101787840*x^13/13! + 16657280640*x^15/15! + 3180450873600*x^17/17! + 837294557299200*x^19/19! +...+ (2*n)!/n! * a(n) * x^(2*n+1)/(2*n+1)! +...
%o (PARI) {a(n) = my(A=1); for(i=1,#binary(n+1), A = exp( intformal( subst(A,x,x^2) +x*O(x^n)) ) ); n!*polcoeff(A,n)}
%o for(n=0,30,print1(a(n),", "))
%Y Cf. A138292.
%K nonn
%O 0,4
%A _Paul D. Hanna_, Nov 05 2017