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
Keywords
Examples
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! +...
Programs
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Mathematica
Join[{1},Table[((n+1)^(n-1)+(n-1)^(n-1))/2,{n,2,30}]] (* Harvey P. Dale, Feb 06 2023 *) -
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),", "))
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).
Extensions
Entry revised by Paul D. Hanna, Jun 19 2016
Corrected and extended by Harvey P. Dale, Feb 06 2023