A196959 E.g.f. satisfies: A(x) = Sum_{n>=0} 1/n! * Sum_{k=0..n} (-1)^(n-k) * C(n,k) * (1 + x*A(x)^(2*k))^k.

1, 1, 9, 193, 6721, 326001, 20316937, 1548374129, 139576777921, 14530808439073, 1715928199384521, 226652340142349793, 33113449456084235905, 5302086923264289694225, 923349950199153833740105, 173761214485224395469845521, 35139709415689684107278235265

Offset

0,3

Links

Table of n, a(n) for n=0..16.

Formula

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

Example

 E.g.f.: A(x) = 1 + x + 9*x^2/2! + 193*x^3/3! + 6721*x^4/4! + 326001*x^5/5! +...
where:
A(x) = 1 + A(x)^2*exp(A(x)^2 - 1)*x + A(x)^8*exp(A(x)^4 - 1)*x^2/2! + A(x)^18*exp(A(x)^6 - 1)*x^3/3! + A(x)^32*exp(A(x)^8 - 1)*x^4/4! +...
Also, e.g.f. A = A(x) satisfies:
A(x) = 1 - (1 - (1+x*A^2)) + 1/2!*(1 - 2*(1+x*A^2) + (1+x*A^4)^2) -
1/3!*(1 - 3*(1+x*A^2) + 3*(1+x*A^4)^2 - (1+x*A^6)^3) +
1/4!*(1 - 4*(1+x*A^2) + 6*(1+x*A^4)^2 - 4*(1+x*A^6)^3 + (1+x*A^8)^4) -
1/5!*(1 - 5*(1+x*A^2) + 10*(1+x*A^4)^2 - 10*(1+x*A^6)^3 + 5*(1+x*A^8)^4 - (1+x*A^10)^5) +-...

Prog

 (PARI) {a(n)=local(A=1+x, X=x+x*O(x^n)); for(i=1, n, A=1+sum(m=1, n, exp(A^(2*m)-1)*A^(2*m^2)*X^m/m!)); n!*polcoeff(A, n)}
(PARI) {a(n)=local(A=1+x, X=x+x*O(x^n)); for(i=1, n, A=1+sum(m=1, n, 1/m!*sum(k=0, m, binomial(m, k)*(-1)^(m-k)*(1+X*A^(2*k))^k))); n!*polcoeff(A, n)}

Crossrefs

Cf. A195947, A196958.

Sequence in context: A189178 A297517 A116877 * A266485 A279132 A334777

Adjacent sequences: A196956 A196957 A196958 * A196960 A196961 A196962

Keyword

nonn

Author

Paul D. Hanna, Oct 08 2011

Status

approved