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A274278 a(n) = ((n+1)^(n-1) - (n-1)^(n-1))/2 for n>=1. 3
1, 0, 1, 6, 49, 520, 6841, 107744, 1979713, 41611392, 985263601, 25958682112, 753424361713, 23888905963520, 821659980883561, 30472793606184960, 1212264580564478209, 51496393511442350080, 2326573297949232710881, 111398795962351731212288, 5635038492335356268228401, 300285949343202022103973888, 16814498551154751682934232601, 987042812055984079330393194496 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

LINKS

Table of n, a(n) for n=0..23.

FORMULA

E.g.f.: cosh(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+1) * n^(n-2*k-2).

EXAMPLE

E.g.f.: A(x) = 1 + x^2/2! + 6*x^3/3! + 49*x^4/4! + 520*x^5/5! + 6841*x^6/6! + 107744*x^7/7! + 1979713*x^8/8! + 41611392*x^9/9! + 985263601*x^10/10! +...

such that A(x) = cosh(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=0, 30, print1(a(n), ", "))

(PARI) {a(n) = sum(k=0, (n-1)\2, binomial(n-1, 2*k+1) * n^(n-2*k-2))}

for(n=0, 30, print1(a(n), ", "))

(PARI) {a(n) = my(W=sum(m=0, n, (m+1)^(m-1)*x^m/m!) +x*O(x^n)); n!*polcoeff(cosh(x*W), n)}

for(n=0, 30, print1(a(n), ", "))

CROSSREFS

Cf. A000272, A195136, A274279.

Sequence in context: A055847 A143165 A008786 * A286799 A245797 A046195

Adjacent sequences:  A274275 A274276 A274277 * A274279 A274280 A274281

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jun 19 2016

STATUS

approved

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Last modified April 11 18:59 EDT 2021. Contains 342888 sequences. (Running on oeis4.)