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A195136 a(n) = ((n+1)^(n-1) + (n-1)^(n-1))/2 for n>=1. 3
1, 2, 10, 76, 776, 9966, 154400, 2803256, 58388608, 1372684090, 35958682112, 1038736032324, 32805006411776, 1124535087475814, 41584800431742976, 1650158470945337584, 69943137585151901696, 3153813559835569475058, 150745204037648268787712, 7613458147995669857352380, 405143549343202022103973888, 405143549343202022103973888, 22657085569540734204315357022 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

LINKS

Table of n, a(n) for n=1..23.

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! +...

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: A066223 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

STATUS

approved

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Last modified April 19 06:59 EDT 2019. Contains 322237 sequences. (Running on oeis4.)