0,2

G.f. satisfies: A(x) = exp( Sum_{n>=1} 3^(n^2)*[x/A(x)]^n/n ).

Let G(x) = exp(Sum_{n>=1} 3^(n^2)*x^n/n) = g.f. of A155203, 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) = A155203(n) ;

(4) [x^n] A(x)^(n+m)*m/(n+m) = [x^n] G(x)^m for |n+m|>0, n>=0.

G.f.: A(x) = 1 + 3*x + 36*x^2 + 6336*x^3 + 10701720*x^4 +...

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

[1,(3), 36, 6336, 10701720, 169328019456, 25013229623639712,...];

[1, 6,(81), 12888, 21442752, 338720705424, 50027476026064896,...];

[1, 9, 135,(19683), 32224068, 508178240640, 75042739500374376,...];

[1, 12, 198, 26748,(43046721), 677700811728, 100059020340421248,...];

[1, 15, 270, 34110, 53911845,(847288609443), 125076318840827460,...];

[1, 18, 351, 41796, 64820655, 1016941828914,(150094635296999121),...];

[1, 21, 441, 49833, 75774447, 1186660669887, 175113970005142539,...];

[1, 24, 540, 58248, 86774598, 1356445336968, 200134323262280988,...];

...

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

The main diagonal = [x^n] A(x)^(n+1) = (n+1)*A155203(n):

[1, 2*3, 3*45, 4*6687, 5*10782369, 6*169490304819, ...].

(PARI) {a(n)=local(G=exp(sum(m=1, n, 3^(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, 3^(m^2)*(x/A)^m/m))); polcoeff(A, n)}

Cf. A164764, A155203.

Sequence in context: A102579 A136393 A158093 this_sequence A088322 A080807 A006268

Adjacent sequences: A163963 A163964 A163965 this_sequence A163967 A163968 A163969

nonn

Paul D. Hanna (pauldhanna(AT)juno.com), Aug 28 2009