0,2

G.f. satisfies: A(x) = exp( Sum_{n>=1} 2^(n^2)*[x/A(x)]^n/n ).

Let G(x) = exp(Sum_{n>=1} 2^(n^2)*x^n/n) = g.f. of A155200, then:

(1) A(x) = G(x/A(x)) and A(x*G(x)) = G(x) ;

(2) A(x) = x/Series_Reversion[x*G(x)] ;

(3) [x^n] A(x)^(n+1)/(n+1) = [x^n] G(x) = A155200(n) ;

(4) [x^n] A(x)^(n+m)*m/(n+m) = [x^n] G(x)^m for all m.

G.f.: A(x) = 1 + 2*x + 6*x^2 + 144*x^3 + 15390*x^4 + 6580224*x^5 +...

The coefficients in the successive powers of A(x) begin:

[1,(2), 6, 144, 15390, 6580224, 11386265292, 80284132772352,...];

[1, 4,(16), 312, 31392, 13223736, 22799056896, 160613894000880,...];

[1, 6, 30,(512), 48078, 19932480, 34238570076, 240989363896320,...];

[1, 8, 48, 752,(65536), 26708592, 45705005568, 321410623205088,...];

[1, 10, 70, 1040, 83870,(33554432), 57198570060, 401877753223680,...];

[1, 12, 96, 1384, 103200, 40472616,(68719476736), 482390835814224,...];

[1, 14, 126, 1792, 123662, 47466048, 80267945884,(562949953421312),...];

[1, 16, 160, 2272, 145408, 54537952, 91844205568, 643555189090240,...];

...

The above terms in parenthesis = [x^n] A(x)^n = 2^(n^2) for n=1,2,3,...

The main diagonal = [x^n] A(x)^(n+1) = (n+1)*A155200(n):

[1, 2*2, 3*10, 4*188, 5*16774, 6*6745436, 7*11466849412, ...].

(PARI) {a(n)=local(G=exp(sum(m=1, n, 2^(m^2)*x^m/m)+x*O(x^n))); polcoeff(x/serreverse(x*G), n)}

(PARI) {a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=exp(sum(m=1, n, 2^(m^2)*(x/A)^m/m))); polcoeff(A, n)}

Cf. A155200, A156214, A156631.

Sequence in context: A047937 A027731 A145143 this_sequence A024397 A015173 A122570

Adjacent sequences: A164761 A164762 A164763 this_sequence A164765 A164766 A164767

nonn

Paul D. Hanna (pauldhanna(AT)juno.com), Aug 25 2009