A294638 E.g.f. satisfies: A'(x) = A(x) * A(x^2).

1, 1, 1, 3, 9, 33, 153, 963, 6129, 47457, 393489, 3689379, 36673209, 410924097, 4810169961, 64694478627, 878318278497, 13230037503297, 203967546446241, 3494178651687363, 60117798742663401, 1137159539308348641, 21683284489630748601, 452680959717183978243, 9454328250188008785489, 214087305044257976127393, 4862802200825123466537393, 119970186740330465448543843, 2944202974922987534742898329

Offset: 0

Author

Paul D. Hanna, Nov 05 2017

Keywords

• nonn

Examples

			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! + ...
such that A'(x) = A(x) * A(x^2).
Also, A(x) = exp( Integral A(x^2) dx ).
RELATED SERIES.
The logarithm of the e.g.f. is an odd function that begins:
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!) +...
which equals Integral A(x^2) dx.
Explicitly,
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)! +...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..520

Crossrefs

Cf. A138292.

Programs

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)}
for(n=0,30,print1(a(n),", "))

Formula

E.g.f. A(x) = Sum_{n>=0} a(n) * x^n/n! satisfies:

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

(2) A(x) = 1/A(-x).

(3) A(x) = exp( Sum_{n>=0} a(n) * x^(2*n+1) / ((2*n+1)*n!) ) .

(4) A(x) = exp( Sum_{n>=0} (2*n)!/n! * a(n) * x^(2*n+1)/(2*n+1)! ).