A163966 G.f. satisfies: the coefficient of x^n in A(x)^n = 3^(n^2) for n>=0.
1, 3, 36, 6336, 10701720, 169328019456, 25013229623639712, 34185168163500076044288, 429209759636992944567490013280, 49269599503729281688381600977015275520
Offset: 0
Keywords
Examples
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, ...].
Programs
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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)}
Formula
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.