login
E.g.f. satisfies: A(x) = Sum_{n>=0} x^n/n! * A(x)^(n(n+1)/2).
5

%I #4 Nov 18 2012 21:36:21

%S 1,1,3,19,191,2656,47392,1034335,26721781,798018616,27058991246,

%T 1027237384009,43172232488959,1990253576425960,99871804451808040,

%U 5419775866582473211,316301430225674131433,19756213549154356027408

%N E.g.f. satisfies: A(x) = Sum_{n>=0} x^n/n! * A(x)^(n(n+1)/2).

%F E.g.f. satisfies: A(x) = B(x*A(x)) and A(x/B(x)) = B(x) where B(x) satisfies:

%F B(x) = Sum_{n>=0} x^n/n! * B(x)^(n*(n-1)/2) and is the e.g.f. of A155804.

%e E.g.f.: A(x) = 1 + x + 3*x^2/2! + 19*x^3/3! + 191*x^4/4! + 2656*x^5/5! +...

%e where e.g.f. A(x) satisfies:

%e A(x) = 1 + x*A(x) + x^2/2!*A(x)^3 + x^3/3!*A(x)^6 + x^4/4!*A(x)^10 +...

%e Let B(x) = A(x/B(x)) be the e.g.f. of A155804 then:

%e B(x) = 1 + x + x^2/2!*B(x) + x^3/3!*B(x)^3 + x^4/4!*B(x)^6 + x^5/5!*B(x)^10 +...

%e B(x) = 1 + x + x^2/2! + 4*x^3/3! + 19*x^4/4! + 161*x^5/5! + 1606*x^6/6! +...

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

%Y Cf. A155804, A155806, A155807, A107591, A219359.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jan 27 2009