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URL: https://oeis.org/A162653

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A162653
E.g.f. satisfies: A(x) = 1 + sinh(x*A(x)).
8
1, 1, 2, 7, 40, 321, 3216, 38263, 530048, 8423425, 151248640, 3026679591, 66766414848, 1609663922881, 42114534017024, 1188556743874711, 35992306103517184, 1164129150556067841, 40053439019074387968
OFFSET
0,3
FORMULA
E.g.f.: A(x) = (1/x)*Series_Reversion[x/(1 + sinh(x))].
a(n) = Sum_{k=0..n} C(n+1,k)/(n+1)*Sum_{j=0..k} (-1)^(k-j)*C(k,j)*(2j-k)^n/2^k.
Let A(x)^m = Sum_{n>=0} a(n,m)*x^n/n!, then
a(n,m) = Sum_{k=0..n} m*C(n+m,k)/(n+m)*Sum_{j=0..k} (-1)^(k-j)*C(k,j)*(2j-k)^n/2^k.
a(n) ~ sqrt(1/r^2 + 1/(r*sqrt(1-r^2))) * n^(n-1) / (exp(n) * r^n), where r = 0.4823099237172185... is the root of the equation cosh(r+sqrt(1-r^2)) = 1/r. - Vaclav Kotesovec, Jan 10 2014
EXAMPLE
E.g.f.: A(x) = 1 + x + 2*x^2/2! + 7*x^3/3! + 40*x^4/4! + 321*x^5/5! +...
MATHEMATICA
CoefficientList[1/x*InverseSeries[Series[x/(1+Sinh[x]), {x, 0, 20}], x], x]*Range[0, 19]! (* Vaclav Kotesovec, Jan 10 2014 *)
PROG
(PARI) {a(n, m=1)=sum(k=0, n, m*binomial(n+m, k)/(n+m)*sum(j=0, k, (-1)^(k-j)*binomial(k, j)*(2*j-k)^n/2^k))}
CROSSREFS
Sequence in context: A308876 A361597 A370878 * A346220 A166904 A105625
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jul 08 2009
STATUS
approved