A162167 E.g.f. satisfies: A(x) = exp(x*exp(2*x*exp(3*x*A(x)))).

1, 1, 5, 61, 945, 18401, 448033, 13162689, 452456833, 17814732769, 790829204481, 39087256226129, 2129136397634689, 126740704552712433, 8186173905423123457, 570241787130949813441, 42616054790603864116737

Offset

0,3

Comments

More generally, if A(x) = exp(p*x*exp(q*x*exp(r*x*A(x)))

where A(x)^m = Sum_{n>=0} a(n,m)*x^n/n!, then

a(n,m) = Sum_{k=0..n} Sum_{j=0..k} p^(n-k)*q^(k-j)*r^j*C(n,k)*C(k,j)*m*(j+m)^(n-k-1)*(n-k)^(k-j)*(k-j)^j.

...

In general, if A(x) = F(x*G(x*H(x*A(x))) with F(0)=G(0)=H(0)=1,

where A(x)^m = Sum_{n>=0} a(n,m)*x^n, then

a(n,m) = Sum_{k=0..n} Sum_{j=0..k} {[x^(n-k)] F(x)^(j+m)*m/(j+m)} * {[x^(k-j)] G(x)^(n-k)} * {[x^j] H(x)^(k-j)}.

...

Links

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

Formula

a(n) = Sum_{k=0..n} Sum_{j=0..k} 2^(k-j)*3^j*C(n,k)*C(k,j)*(j+1)^(n-k-1)*(n-k)^(k-j)*(k-j)^j.

Example

E.g.f.: A(x) = 1 + x + 5*x^2/2! + 61*x^3/3! + 945*x^4/4! + 18401*x^5/5! +...

Prog

(PARI) {a(n, m=1, p=1, q=2, r=3)=n!*sum(k=0, n, sum(j=0, k, p^(n-k)*q^(k-j)*r^j*m*(j+m)^(n-k-1)/(n-k)!*(n-k)^(k-j)/(k-j)!*(k-j)^j/j!))}

Crossrefs

Cf. A162161, A162162.

Sequence in context: A363867 A012060 A012167 * A134282 A146760 A294024

Adjacent sequences: A162164 A162165 A162166 * A162168 A162169 A162170

Keyword

nonn

Author

Paul D. Hanna, Jun 27 2009

Extensions

Paul D. Hanna, Jun 28 2009

Status

approved