A195136 - OEIS

%I #21 Feb 06 2023 12:31:53

%S 1,2,10,76,776,9966,154400,2803256,58388608,1372684090,35958682112,

%T 1038736032324,32805006411776,1124535087475814,41584800431742976,

%U 1650158470945337584,69943137585151901696,3153813559835569475058,150745204037648268787712,7613458147995669857352380,405143549343202022103973888,22657085569540734204315357022,1328470689420203636727039918080,81494507575933974604289943213096,5220210773193749540624447754469376,348542314841685116176787263033063466,24216786265392720787141148530274467840,1748280517106781152846793195054531026356,130956723831431687431286364126682302906368,10164786953127554557192799138093559445158870

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

%F 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!.

%F a(n) = Sum_{k=0..floor((n-1)/2)} C(n-1,2*k) * n^(n-2*k-1).

%e 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! +...

%e such that A(x) = sinh(x*W(x))

%e where W(x) = LambertW(-x)/(-x) begins

%e 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! +...

%e and satisfies W(x) = exp(x*W(x)).

%e Also, A(x) = (W(x) - 1/W(x))/2 where

%e 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! +...

%t Join[{1},Table[((n+1)^(n-1)+(n-1)^(n-1))/2,{n,2,30}]] (* _Harvey P. Dale_, Feb 06 2023 *)

%o (PARI) {a(n)=((n+1)^(n-1) + (n-1)^(n-1))/2}

%o for(n=1,30,print1(a(n),", "))

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

%o for(n=1,30,print1(a(n),", "))

%o (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)}

%o for(n=1,30,print1(a(n),", "))

%Y Cf. A000272, A274278, A274279.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Sep 09 2011

%E Entry revised by _Paul D. Hanna_, Jun 19 2016

%E Corrected and extended by _Harvey P. Dale_, Feb 06 2023