

A136647


G.f.: A(x) = Sum_{n>=0} asinh( 2^n*x )^n / n! ; a power series in x with integer coefficients.


2



1, 2, 8, 84, 2688, 276892, 94978048, 111457917800, 457117679616000, 6660816097416169260, 349290546231751288553472, 66597307693046550483175282456, 46556113319179632622352835689840640
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OFFSET

0,2


LINKS

Table of n, a(n) for n=0..12.


FORMULA

a(n) = [y^n] ( sqrt(1+y^2) + y )^(2^n), since log(sqrt(1+y^2) + y) = asinh(y); [y^n] F(y) denotes the coefficient of y^n in F(y).


EXAMPLE

G.f.: A(x) = 1 + 2*x + 8*x^2 + 84*x^3 + 2688*x^4 + 276892*x^5 +...
This is a special application of the following identity.
Let F(x),G(x), be power series in x such that F(0)=1,G(0)=1, then
Sum_{n>=0} m^n * H(q^n*x) * log( F(q^n*x)*G(x) )^n / n! =
Sum_{n>=0} x^n * G(x)^(m*q^n) * [y^n] H(y)*F(y)^(m*q^n).


PROG

(PARI) {a(n)=polcoeff(sum(k=0, n, asinh(2^k*x +x*O(x^n))^k/k!), n)} (PARI) {a(n)=polcoeff((sqrt(1+x^2)+x+x*O(x^n))^(2^n), n)}


CROSSREFS

Cf. A136558.
Sequence in context: A013175 A120820 A134089 * A052456 A000532 A083831
Adjacent sequences: A136644 A136645 A136646 * A136648 A136649 A136650


KEYWORD

nonn


AUTHOR

Paul D. Hanna, Jan 20 2008


STATUS

approved



