A195136 a(n) = ((n+1)^(n-1) + (n-1)^(n-1))/2 for n>=1.

1, 2, 10, 76, 776, 9966, 154400, 2803256, 58388608, 1372684090, 35958682112, 1038736032324, 32805006411776, 1124535087475814, 41584800431742976, 1650158470945337584, 69943137585151901696, 3153813559835569475058, 150745204037648268787712, 7613458147995669857352380, 405143549343202022103973888, 22657085569540734204315357022, 1328470689420203636727039918080, 81494507575933974604289943213096, 5220210773193749540624447754469376, 348542314841685116176787263033063466, 24216786265392720787141148530274467840, 1748280517106781152846793195054531026356, 130956723831431687431286364126682302906368, 10164786953127554557192799138093559445158870

Offset

1,2

Links

Table of n, a(n) for n=1..30.

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

Cf. A000272, A274278, A274279.

Sequence in context: A355110 A088500 A295929 * A294573 A301741 A140763

Adjacent sequences: A195133 A195134 A195135 * A195137 A195138 A195139

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