OFFSET
1,2
FORMULA
E.g.f.: sinh(x*W(x)) = (W(x) - 1/W(x))/2 where W(x) = LambertW(-x)/(-x) = exp(x*W(x)) = Sum_{n>=0} (n+1)^(n-1)*x^n/n!.
a(n) = Sum_{k=0..floor((n-1)/2)} C(n-1,2*k) * n^(n-2*k-1).
EXAMPLE
E.g.f.: A(x) = x + 2*x^2/2! + 10*x^3/3! + 76*x^4/4! + 776*x^5/5! + 9966*x^6/6! + 154400*x^7/7! + 2803256*x^8/8! + 58388608*x^9/9! + 1372684090*x^10/10! +...
such that A(x) = sinh(x*W(x))
where W(x) = LambertW(-x)/(-x) begins
W(x) = 1 + x + 3*x^2/2! + 16*x^3/3! + 125*x^4/4! + 1296*x^5/5! + 16807*x^6/6! + 262144*x^7/7! + 4782969*x^8/8! + 100000000*x^9/9! +...+ (n+1)^(n-1)*x^n/n! +...
and satisfies W(x) = exp(x*W(x)).
Also, A(x) = (W(x) - 1/W(x))/2 where
1/W(x) = 1 - x - x^2/2! - 4*x^3/3! - 27*x^4/4! - 256*x^5/5! - 3125*x^6/6! - 46656*x^7/7! - 823543*x^8/8! +...+ -(n-1)^(n-1)*x^n/n! +...
MATHEMATICA
Join[{1}, Table[((n+1)^(n-1)+(n-1)^(n-1))/2, {n, 2, 30}]] (* Harvey P. Dale, Feb 06 2023 *)
PROG
(PARI) {a(n)=((n+1)^(n-1) + (n-1)^(n-1))/2}
for(n=1, 30, print1(a(n), ", "))
(PARI) {a(n)=sum(k=0, (n-1)\2, binomial(n-1, 2*k)*n^(n-2*k-1))}
for(n=1, 30, print1(a(n), ", "))
(PARI) {a(n)=local(W=sum(m=0, n, (m+1)^(m-1)*x^m/m!)+x*O(x^n)); n!*polcoeff(sinh(x*W), n)}
for(n=1, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Sep 09 2011
EXTENSIONS
Entry revised by Paul D. Hanna, Jun 19 2016
Corrected and extended by Harvey P. Dale, Feb 06 2023
STATUS
approved