A161568 E.g.f. satisfies: A(x) = exp(2*x*exp(3*x*A(x))).

1, 2, 16, 206, 3976, 101402, 3237220, 124293206, 5582747824, 287346080690, 16680250440124, 1078327289938670, 76840445565238024, 5984507179839282122, 505778795448930860308, 46104043794638089809158

Offset

0,2

Links

G. C. Greubel, Table of n, a(n) for n = 0..341

Formula

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

More generally, if G(x) = exp(p*x*exp(q*x*G(x))) = Sum_{n>=0} g(n)*x^n/n!,

then g(n) = Sum_{k=0..n} C(n,k) * p^k * q^(n-k) * (n-k+1)^(k-1) * k^(n-k).

a(n) ~ sqrt(s/3) * n^(n-1) / (exp(n) * r^(n+1/2)), where r = 0.149417197143691584817... and s = 2.468671804906329807069... are roots of the system of equations 3*r*s*Log(s) = 1, 6*exp(3*r*s)*s*r^2 = 1. - Vaclav Kotesovec, Jul 15 2014

Example

E.g.f.: A(x) = 1 + 2*x + 16*x^2/2! + 206*x^3/3! + 3976*x^4/4! +...

Mathematica

Flatten[{1, Table[Sum[Binomial[n, k] * 2^k * 3^(n-k) * (n-k+1)^(k-1) * k^(n-k), {k, 0, n}], {n, 1, 20}]}] (* Vaclav Kotesovec, Jul 15 2014 *)

Prog

(PARI) {a(n)=sum(k=0, n, binomial(n, k)*2^k*3^(n-k)*(n-k+1)^(k-1)*k^(n-k))}
(PARI) {a(n)=local(A=1+x); for(i=0, n, A=exp(2*x*exp(3*x*A+O(x^n)))); n!*polcoeff(A, n, x)}

Crossrefs

Cf. A161565, A161566, A161567.

Sequence in context: A303673 A367425 A360466 * A138429 A087923 A222523

Adjacent sequences: A161565 A161566 A161567 * A161569 A161570 A161571

Keyword

nonn

Author

Paul D. Hanna, Jun 14 2009

Status

approved